{"id":530849,"date":"2026-03-13T15:44:22","date_gmt":"2026-03-13T08:44:22","guid":{"rendered":"https:\/\/hoanghamobile.com\/tin-tuc\/?p=530849"},"modified":"2026-03-13T15:44:22","modified_gmt":"2026-03-13T08:44:22","slug":"dao-ham-e-mu-u-la-gi-cong-thuc-cach-tinh-va-bai-tap-ung-dung-chi-tiet-nhat","status":"publish","type":"post","link":"https:\/\/hoanghamobile.com\/tin-tuc\/dao-ham-e-mu-u-la-gi-cong-thuc-cach-tinh-va-bai-tap-ung-dung-chi-tiet-nhat\/","title":{"rendered":"\u0110\u1ea1o h\u00e0m e m\u0169 u l\u00e0 g\u00ec? C\u00f4ng th\u1ee9c, c\u00e1ch t\u00ednh v\u00e0 b\u00e0i t\u1eadp \u1ee9ng d\u1ee5ng chi ti\u1ebft nh\u1ea5t"},"content":{"rendered":"<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">T\u1eeb c\u00e1c b\u00e0i to\u00e1n t\u0103ng tr\u01b0\u1edfng, m\u00f4 h\u00ecnh v\u1eadt l\u00fd, \u0111\u1ebfn ph\u00e2n t\u00edch t\u00e0i ch\u00ednh, h\u00e0m s\u1ed1 m\u0169 xu\u1ea5t hi\u1ec7n \u1edf kh\u1eafp m\u1ecdi n\u01a1i. Trong \u0111\u00f3, <\/span><b>\u0111\u1ea1o h\u00e0m e m\u0169 u<\/b><span style=\"font-weight: 400;\"> l\u00e0 m\u1ed9t c\u00f4ng th\u1ee9c c\u01a1 b\u1ea3n nh\u01b0ng c\u1ef1c k\u1ef3 quan tr\u1ecdng, gi\u00fap ta ph\u00e2n t\u00edch t\u1ed1c \u0111\u1ed9 bi\u1ebfn thi\u00ean c\u1ee7a c\u00e1c qu\u00e1 tr\u00ecnh t\u1ef1 nhi\u00ean v\u00e0 x\u00e3 h\u1ed9i. B\u00e0i vi\u1ebft n\u00e0y s\u1ebd gi\u00fap b\u1ea1n n\u1eafm v\u1eefng \u0111\u1ecbnh ngh\u0129a, c\u00f4ng th\u1ee9c, c\u00e1ch t\u00ednh, v\u00ed d\u1ee5 minh h\u1ecda v\u00e0 \u1ee9ng d\u1ee5ng th\u1ef1c ti\u1ec5n c\u1ee7a \u0111\u1ea1o h\u00e0m \u2013 t\u1eeb l\u00fd thuy\u1ebft c\u01a1 b\u1ea3n \u0111\u1ebfn c\u00e1c tr\u01b0\u1eddng h\u1ee3p n\u00e2ng cao, k\u00e8m l\u1eddi gi\u1ea3i chi ti\u1ebft cho t\u1eebng d\u1ea1ng b\u00e0i t\u1eadp ph\u1ed5 bi\u1ebfn.<\/span><\/p>\n<h2 style=\"text-align: justify;\"><b>T\u1ed5ng quan v\u1ec1 h\u00e0m s\u1ed1 m\u0169 t\u1ef1 nhi\u00ean e v\u00e0 e m\u0169 x<\/b><\/h2>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">S\u1ed1 <\/span><span style=\"font-weight: 400;\">e<\/span><span style=\"font-weight: 400;\"> l\u00e0 m\u1ed9t h\u1eb1ng s\u1ed1 to\u00e1n h\u1ecdc \u0111\u1eb7c bi\u1ec7t, c\u00f3 gi\u00e1 tr\u1ecb x\u1ea5p x\u1ec9 2,71828\u2026<\/span><span style=\"font-weight: 400;\">, \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 c\u01a1 s\u1ed1 t\u1ef1 nhi\u00ean. H\u00e0m s\u1ed1 m\u0169 v\u1edbi c\u01a1 s\u1ed1 <\/span><span style=\"font-weight: 400;\">e<\/span><span style=\"font-weight: 400;\"> c\u00f3 vai tr\u00f2 quan tr\u1ecdng trong nhi\u1ec1u l\u0129nh v\u1ef1c v\u00ec t\u00ednh ch\u1ea5t \u0111\u1ed9c \u0111\u00e1o c\u1ee7a n\u00f3:<\/span><\/p>\n<ul style=\"text-align: justify;\">\n<li style=\"font-weight: 400;\"><b>e m\u0169 x l\u00e0 g\u00ec?<\/b> <span style=\"font-weight: 400;\">e^x<\/span><span style=\"font-weight: 400;\"> l\u00e0 h\u00e0m s\u1ed1 m\u0169 v\u1edbi c\u01a1 s\u1ed1 <\/span><span style=\"font-weight: 400;\">e<\/span><span style=\"font-weight: 400;\">, bi\u1ebfn s\u1ed1 l\u00e0 <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">. \u0110\u1eb7c bi\u1ec7t, \u0111\u1ea1o h\u00e0m c\u1ee7a <\/span><span style=\"font-weight: 400;\">e^x<\/span><span style=\"font-weight: 400;\"> l\u1ea1i ch\u00ednh l\u00e0 <\/span><span style=\"font-weight: 400;\">e^x<\/span><span style=\"font-weight: 400;\">, t\u1ea1o n\u00ean s\u1ef1 \u0111\u01a1n gi\u1ea3n trong t\u00ednh to\u00e1n.<\/span><\/li>\n<li style=\"font-weight: 400;\"><b>H\u00e0m s\u1ed1 e^u:<\/b><span style=\"font-weight: 400;\"> Khi <\/span><span style=\"font-weight: 400;\">u<\/span><span style=\"font-weight: 400;\"> l\u00e0 m\u1ed9t h\u00e0m s\u1ed1 b\u1ea5t k\u1ef3 theo <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\"> (v\u00ed d\u1ee5: <\/span><span style=\"font-weight: 400;\">u = ax + b<\/span><span style=\"font-weight: 400;\">, <\/span><span style=\"font-weight: 400;\">u = x^2<\/span><span style=\"font-weight: 400;\">, <\/span><span style=\"font-weight: 400;\">u = \\sin x<\/span><span style=\"font-weight: 400;\">\u2026), ta c\u00f3 h\u00e0m h\u1ee3p <\/span><span style=\"font-weight: 400;\">e^u<\/span><span style=\"font-weight: 400;\">. Vi\u1ec7c l\u1ea5y \u0111\u1ea1o h\u00e0m h\u00e0m h\u1ee3p n\u00e0y gi\u00fap ta ph\u00e2n t\u00edch c\u00e1c bi\u1ebfn \u0111\u1ed5i ph\u1ee9c t\u1ea1p trong th\u1ef1c t\u1ebf.<\/span><\/li>\n<li style=\"font-weight: 400;\"><b>Li\u00ean h\u1ec7 gi\u1eefa e^x v\u00e0 e^u:<\/b> <span style=\"font-weight: 400;\">e^x<\/span><span style=\"font-weight: 400;\"> l\u00e0 tr\u01b0\u1eddng h\u1ee3p \u0111\u1eb7c bi\u1ec7t c\u1ee7a <\/span><span style=\"font-weight: 400;\">e^u<\/span><span style=\"font-weight: 400;\"> khi <\/span><span style=\"font-weight: 400;\">u = x<\/span><span style=\"font-weight: 400;\">. C\u00f4ng th\u1ee9c \u0111\u1ea1o h\u00e0m t\u1ed5ng qu\u00e1t cho <\/span><span style=\"font-weight: 400;\">e^u<\/span><span style=\"font-weight: 400;\"> s\u1ebd bao g\u1ed3m th\u00eam h\u1ec7 s\u1ed1 <\/span><span style=\"font-weight: 400;\">u&#8217;<\/span><span style=\"font-weight: 400;\"> (\u0111\u1ea1o h\u00e0m c\u1ee7a u theo x).<\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\"><picture><source srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/webp-express\/webp-images\/uploads\/2026\/03\/dao-ham-e-mu-u-1.jpg.webp 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/webp-express\/webp-images\/uploads\/2026\/03\/dao-ham-e-mu-u-1-300x169.jpg.webp 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/webp-express\/webp-images\/uploads\/2026\/03\/dao-ham-e-mu-u-1-768x432.jpg.webp 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" type=\"image\/webp\"><img fetchpriority=\"high\" decoding=\"async\" class=\"aligncenter size-full wp-image-530923 webpexpress-processed\" src=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/03\/dao-ham-e-mu-u-1.jpg\" alt=\"dao-ham-e-mu-u-1\" width=\"800\" height=\"450\" srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/03\/dao-ham-e-mu-u-1.jpg 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/03\/dao-ham-e-mu-u-1-300x169.jpg 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/03\/dao-ham-e-mu-u-1-768x432.jpg 768w\" sizes=\"(max-width: 800px) 100vw, 800px\"><\/picture><\/p>\n<h2 style=\"text-align: justify;\"><b>C\u00f4ng th\u1ee9c \u0111\u1ea1o h\u00e0m e m\u0169 u t\u1ed5ng qu\u00e1t<\/b><\/h2>\n<p style=\"text-align: justify;\"><b>C\u00f4ng th\u1ee9c t\u1ed5ng qu\u00e1t:<\/b><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">(e^u)&#8217; = e^u \\cdot u&#8217;<\/span><\/p>\n<p style=\"text-align: justify;\"><b>Gi\u1ea3i th\u00edch c\u00f4ng th\u1ee9c (e^u)&#8217; = e^u \u00d7 u&#8217;<\/b><\/p>\n<ul style=\"text-align: justify;\">\n<li style=\"font-weight: 400;\"><b>e^u:<\/b><span style=\"font-weight: 400;\"> L\u00e0 h\u00e0m s\u1ed1 m\u0169 v\u1edbi bi\u1ec3u th\u1ee9c <\/span><span style=\"font-weight: 400;\">u = u(x)<\/span><span style=\"font-weight: 400;\"> t\u00f9y \u00fd.<\/span><\/li>\n<li style=\"font-weight: 400;\"><b>u&#8217;:<\/b><span style=\"font-weight: 400;\"> L\u00e0 \u0111\u1ea1o h\u00e0m c\u1ee7a <\/span><span style=\"font-weight: 400;\">u<\/span><span style=\"font-weight: 400;\"> theo bi\u1ebfn <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">, k\u00fd hi\u1ec7u <\/span><span style=\"font-weight: 400;\">\\frac{du}{dx}<\/span><span style=\"font-weight: 400;\"> ho\u1eb7c <\/span><span style=\"font-weight: 400;\">u'(x)<\/span><span style=\"font-weight: 400;\">.<\/span><\/li>\n<li style=\"font-weight: 400;\"><b>\u00dd ngh\u0129a:<\/b><span style=\"font-weight: 400;\"> Khi l\u1ea5y \u0111\u1ea1o h\u00e0m h\u00e0m h\u1ee3p, ta ph\u1ea3i nh\u00e2n th\u00eam \u0111\u1ea1o h\u00e0m ph\u1ea7n &#8220;b\u00ean trong&#8221; \u2013 g\u1ecdi l\u00e0 <\/span><b>quy t\u1eafc d\u00e2y chuy\u1ec1n<\/b><span style=\"font-weight: 400;\"> (chain rule).<\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\"><b>S\u1ef1 kh\u00e1c bi\u1ec7t v\u1edbi \u0111\u1ea1o h\u00e0m e m\u0169 x:<\/b><span style=\"font-weight: 400;\"> N\u1ebfu <\/span><span style=\"font-weight: 400;\">u = x<\/span><span style=\"font-weight: 400;\">, ta c\u00f3 <\/span><span style=\"font-weight: 400;\">u&#8217; = 1<\/span><span style=\"font-weight: 400;\">, do \u0111\u00f3 c\u00f4ng th\u1ee9c tr\u1edf th\u00e0nh <\/span><span style=\"font-weight: 400;\">(e^x)&#8217; = e^x \\cdot 1 = e^x<\/span><span style=\"font-weight: 400;\">. V\u1edbi c\u00e1c <\/span><span style=\"font-weight: 400;\">u(x)<\/span><span style=\"font-weight: 400;\"> ph\u1ee9c t\u1ea1p h\u01a1n, c\u1ea7n nh\u00e2n th\u00eam <\/span><span style=\"font-weight: 400;\">u&#8217;<\/span><span style=\"font-weight: 400;\">.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">Lu\u00f4n nh\u1edb:<\/span><span style=\"font-weight: 400;\"> \u0110\u1ea1o h\u00e0m c\u1ee7a <\/span><span style=\"font-weight: 400;\">e^u<\/span><span style=\"font-weight: 400;\"> b\u1eb1ng <\/span><span style=\"font-weight: 400;\">e^u<\/span><span style=\"font-weight: 400;\"> nh\u00e2n v\u1edbi \u0111\u1ea1o h\u00e0m c\u1ee7a <\/span><span style=\"font-weight: 400;\">u<\/span><span style=\"font-weight: 400;\">.<\/span><\/p>\n<h2 style=\"text-align: justify;\"><b>C\u00e1ch t\u00ednh \u0111\u1ea1o h\u00e0m e m\u0169 u t\u1eebng b\u01b0\u1edbc<\/b><\/h2>\n<p style=\"text-align: justify;\"><b>\u0110\u1ec3 t\u00ednh \u0111\u1ea1o h\u00e0m c\u1ee7a e m\u0169 u, b\u1ea1n ch\u1ec9 c\u1ea7n \u00e1p d\u1ee5ng c\u00f4ng th\u1ee9c t\u1ed5ng qu\u00e1t:<\/b><span style=\"font-weight: 400;\"> L\u1ea5y \u0111\u1ea1o h\u00e0m c\u1ee7a u, sau \u0111\u00f3 nh\u00e2n v\u1edbi e^u.<\/span><\/p>\n<ol style=\"text-align: justify;\">\n<li style=\"font-weight: 400;\"><b>B\u01b0\u1edbc 1:<\/b><span style=\"font-weight: 400;\"> X\u00e1c \u0111\u1ecbnh bi\u1ec3u th\u1ee9c <\/span><span style=\"font-weight: 400;\">u(x)<\/span><span style=\"font-weight: 400;\"> b\u00ean trong s\u1ed1 m\u0169.<\/span><\/li>\n<li style=\"font-weight: 400;\"><b>B\u01b0\u1edbc 2:<\/b><span style=\"font-weight: 400;\"> T\u00ednh \u0111\u1ea1o h\u00e0m <\/span><span style=\"font-weight: 400;\">u'(x)<\/span><span style=\"font-weight: 400;\"> theo bi\u1ebfn s\u1ed1 (th\u01b0\u1eddng l\u00e0 <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">).<\/span><\/li>\n<li style=\"font-weight: 400;\"><b>B\u01b0\u1edbc 3:<\/b><span style=\"font-weight: 400;\"> \u00c1p d\u1ee5ng c\u00f4ng th\u1ee9c: <\/span><span style=\"font-weight: 400;\">(e^{u(x)})&#8217; = e^{u(x)} \\cdot u'(x)<\/span><span style=\"font-weight: 400;\">.<\/span><\/li>\n<li style=\"font-weight: 400;\"><b>B\u01b0\u1edbc 4:<\/b><span style=\"font-weight: 400;\"> K\u1ebft h\u1ee3p v\u00e0 r\u00fat g\u1ecdn bi\u1ec3u th\u1ee9c n\u1ebfu c\u1ea7n.<\/span><\/li>\n<\/ol>\n<p style=\"text-align: justify;\"><b>V\u00ed d\u1ee5 1:<\/b><span style=\"font-weight: 400;\"> T\u00ednh \u0111\u1ea1o h\u00e0m <\/span><span style=\"font-weight: 400;\">y = e^{3x+2}<\/span><span style=\"font-weight: 400;\">.<\/span><\/p>\n<p style=\"text-align: justify;\"><b>Gi\u1ea3i:<\/b><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">u = 3x+2 \\rightarrow u&#8217; = 3<\/span><\/p>\n<p style=\"text-align: justify;\"><b>\u00c1p d\u1ee5ng c\u00f4ng th\u1ee9c:<\/b> <span style=\"font-weight: 400;\">y&#8217; = e^{3x+2} \\cdot 3 = 3e^{3x+2}<\/span><\/p>\n<p style=\"text-align: justify;\"><b>V\u00ed d\u1ee5 2:<\/b><span style=\"font-weight: 400;\"> T\u00ednh \u0111\u1ea1o h\u00e0m <\/span><span style=\"font-weight: 400;\">y = e^{\\sin x}<\/span><span style=\"font-weight: 400;\">.<\/span><\/p>\n<p style=\"text-align: justify;\"><b>Gi\u1ea3i:<\/b><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">u = \\sin x \\rightarrow u&#8217; = \\cos x<\/span><\/p>\n<p style=\"text-align: justify;\"><b>\u00c1p d\u1ee5ng c\u00f4ng th\u1ee9c:<\/b> <span style=\"font-weight: 400;\">y&#8217; = e^{\\sin x} \\cdot \\cos x<\/span><\/p>\n<p><picture><source srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/webp-express\/webp-images\/uploads\/2026\/03\/dao-ham-e-mu-u-2.jpg.webp 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/webp-express\/webp-images\/uploads\/2026\/03\/dao-ham-e-mu-u-2-300x169.jpg.webp 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/webp-express\/webp-images\/uploads\/2026\/03\/dao-ham-e-mu-u-2-768x432.jpg.webp 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" type=\"image\/webp\"><img decoding=\"async\" class=\"aligncenter size-full wp-image-530924 webpexpress-processed\" src=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/03\/dao-ham-e-mu-u-2.jpg\" alt=\"dao-ham-e-mu-u-2\" width=\"800\" height=\"450\" srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/03\/dao-ham-e-mu-u-2.jpg 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/03\/dao-ham-e-mu-u-2-300x169.jpg 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/03\/dao-ham-e-mu-u-2-768x432.jpg 768w\" sizes=\"(max-width: 800px) 100vw, 800px\"><\/picture><\/p>\n<h2><strong>C\u00e1c tr\u01b0\u1eddng h\u1ee3p \u0111\u1ea1o h\u00e0m e m\u0169 u<\/strong><\/h2>\n<p style=\"text-align: justify;\"><b>\u0110\u1ea1o h\u00e0m c\u1ee7a e m\u0169 u<\/b><span style=\"font-weight: 400;\"> c\u00f3 nhi\u1ec1u tr\u01b0\u1eddng h\u1ee3p \u0111\u1eb7c bi\u1ec7t t\u00f9y v\u00e0o bi\u1ec3u th\u1ee9c u:<\/span><\/p>\n<p style=\"text-align: justify;\"><b>\u0110\u1ea1o h\u00e0m e m\u0169 x<\/b><\/p>\n<p><b>V\u1edbi u = x:<\/b> <span style=\"font-weight: 400;\">(e^x)&#8217; = e^x \\cdot 1 = e^x<\/span><\/p>\n<p style=\"text-align: justify;\"><b>\u0110\u1ea1o h\u00e0m e m\u0169 tr\u1eeb x<\/b><\/p>\n<p><b>V\u1edbi u = -x:<\/b> <span style=\"font-weight: 400;\">(e^{-x})&#8217; = e^{-x} \\cdot (-1) = -e^{-x}<\/span><b><\/b><\/p>\n<p style=\"text-align: justify;\"><b>\u0110\u1ea1o h\u00e0m e m\u0169 ax+b<\/b><\/p>\n<p><b>V\u1edbi u = ax+b:<\/b> <span style=\"font-weight: 400;\">(e^{ax+b})&#8217; = e^{ax+b} \\cdot a = a e^{ax+b}<\/span><b><\/b><\/p>\n<p style=\"text-align: justify;\"><b>\u0110\u1ea1o h\u00e0m e m\u0169 x b\u00ecnh ph\u01b0\u01a1ng<\/b><\/p>\n<p style=\"text-align: justify;\"><b>V\u1edbi u = x^2:<\/b> <span style=\"font-weight: 400;\">(e^{x^2})&#8217; = e^{x^2} \\cdot 2x = 2x e^{x^2}<\/span><\/p>\n<p style=\"text-align: justify;\"><b>\u0110\u1ea1o h\u00e0m e m\u0169 h\u00e0m l\u01b0\u1ee3ng gi\u00e1c<\/b><\/p>\n<ul style=\"text-align: justify;\">\n<li style=\"font-weight: 400;\"><b>u = sin x: <\/b><span style=\"font-weight: 400;\">(e^{\\sin x})&#8217; = e^{\\sin x} \\cdot \\cos x<\/span><\/li>\n<li style=\"font-weight: 400;\"><b>u = tan x: <\/b><span style=\"font-weight: 400;\">(e^{\\tan x})&#8217; = e^{\\tan x} \\cdot (1+\\tan^2 x)<\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\"><b>Tr\u01b0\u1eddng h\u1ee3p t\u1ed5ng qu\u00e1t v\u00e0 v\u00ed d\u1ee5 n\u00e2ng cao<\/b><\/p>\n<p style=\"text-align: justify;\"><b>V\u00ed d\u1ee5:<\/b><span style=\"font-weight: 400;\"> T\u00ednh \u0111\u1ea1o h\u00e0m <\/span><span style=\"font-weight: 400;\">y = e^{x^3 + 2x}<\/span><span style=\"font-weight: 400;\">.<\/span><\/p>\n<p style=\"text-align: justify;\"><b>Gi\u1ea3i:<\/b> <span style=\"font-weight: 400;\">u = x^3 + 2x \\rightarrow u&#8217; = 3x^2 + 2<\/span><\/p>\n<p style=\"text-align: justify;\"><b>\u00c1p d\u1ee5ng c\u00f4ng th\u1ee9c:<\/b> <span style=\"font-weight: 400;\">y&#8217; = e^{x^3 + 2x} \\cdot (3x^2 + 2)<\/span><\/p>\n<h2 style=\"text-align: justify;\"><b>Ph\u00e2n bi\u1ec7t \u0111\u1ea1o h\u00e0m v\u00e0 nguy\u00ean h\u00e0m li\u00ean quan \u0111\u1ebfn e m\u0169 u<\/b><\/h2>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">\u0110\u1ea1o h\u00e0m v\u00e0 nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 m\u0169 <\/span><span style=\"font-weight: 400;\">e^u<\/span><span style=\"font-weight: 400;\"> l\u00e0 hai kh\u00e1i ni\u1ec7m tr\u00e1i ng\u01b0\u1ee3c: \u0111\u1ea1o h\u00e0m \u0111o t\u1ed1c \u0111\u1ed9 thay \u0111\u1ed5i, nguy\u00ean h\u00e0m t\u00ecm ra h\u00e0m g\u1ed1c.<\/span><\/p>\n<ul style=\"text-align: justify;\">\n<li style=\"font-weight: 400;\"><b>\u0110\u1ea1o h\u00e0m e m\u0169 u:<\/b> <span style=\"font-weight: 400;\">(e^u)&#8217; = e^u \\cdot u&#8217;<\/span><\/li>\n<li style=\"font-weight: 400;\"><b>Nguy\u00ean h\u00e0m e m\u0169 u:<\/b> <span style=\"font-weight: 400;\">\\int e^u \\, du = e^u + C<\/span><span style=\"font-weight: 400;\"> ho\u1eb7c <\/span><span style=\"font-weight: 400;\">\\int e^{u(x)} dx = e^{u(x)} \/ u'(x) + C<\/span><span style=\"font-weight: 400;\"> (n\u1ebfu \u0111\u1ed5i bi\u1ebfn).<\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\"><b>V\u00ed d\u1ee5:<\/b><\/p>\n<ul style=\"text-align: justify;\">\n<li style=\"font-weight: 400;\"><b>\u0110\u1ea1o h\u00e0m:<\/b> <span style=\"font-weight: 400;\">y = e^{2x} \\rightarrow y&#8217; = 2e^{2x}<\/span><\/li>\n<li style=\"font-weight: 400;\"><b>Nguy\u00ean h\u00e0m:<\/b> <span style=\"font-weight: 400;\">\\int e^{2x} dx = \\frac{1}{2}e^{2x} + C<\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\"><b>L\u01b0u \u00fd:<\/b><span style=\"font-weight: 400;\"> \u0110\u1ea1o h\u00e0m <\/span><span style=\"font-weight: 400;\">e^u<\/span><span style=\"font-weight: 400;\"> nh\u00e2n th\u00eam <\/span><span style=\"font-weight: 400;\">u&#8217;<\/span><span style=\"font-weight: 400;\">, c\u00f2n nguy\u00ean h\u00e0m <\/span><span style=\"font-weight: 400;\">e^u<\/span><span style=\"font-weight: 400;\"> ph\u1ea3i chia cho <\/span><span style=\"font-weight: 400;\">u&#8217;<\/span><span style=\"font-weight: 400;\"> (n\u1ebfu l\u1ea5y theo bi\u1ebfn kh\u00e1c).<\/span><\/p>\n<table>\n<tbody>\n<tr>\n<td><b>\u0110\u1ea1o h\u00e0m<\/b><\/td>\n<td><b>Nguy\u00ean h\u00e0m<\/b><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">(e^u)&#8217; = e^u \u00b7 u&#8217;<\/span><\/td>\n<td><span style=\"font-weight: 400;\">\u222be^u du = e^u + C<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">V\u00ed d\u1ee5: (e^{3x})&#8217; = 3e^{3x}<\/span><\/td>\n<td><span style=\"font-weight: 400;\">\u222be^{3x} dx = (1\/3)e^{3x} + C<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">Vi\u1ec7c ph\u00e2n bi\u1ec7t hai kh\u00e1i ni\u1ec7m n\u00e0y gi\u00fap b\u1ea1n tr\u00e1nh nh\u1ea7m l\u1eabn khi gi\u1ea3i to\u00e1n ho\u1eb7c \u00e1p d\u1ee5ng v\u00e0o th\u1ef1c ti\u1ec5n.<\/span><\/p>\n<h2 style=\"text-align: justify;\"><b>B\u00e0i t\u1eadp \u0111\u1ea1o h\u00e0m e m\u0169 u c\u00f3 l\u1eddi gi\u1ea3i<\/b><\/h2>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">D\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t s\u1ed1 b\u00e0i t\u1eadp v\u1ec1 \u0111\u1ea1o h\u00e0m c\u1ee7a e m\u0169 u t\u1eeb c\u01a1 b\u1ea3n \u0111\u1ebfn n\u00e2ng cao, k\u00e8m l\u1eddi gi\u1ea3i chi ti\u1ebft \u0111\u1ec3 b\u1ea1n luy\u1ec7n t\u1eadp.<\/span><\/p>\n<p style=\"text-align: justify;\"><b>B\u00e0i 1:<\/b><span style=\"font-weight: 400;\"> T\u00ednh \u0111\u1ea1o h\u00e0m <\/span><span style=\"font-weight: 400;\">y = e^{5x}<\/span><span style=\"font-weight: 400;\">.<\/span><\/p>\n<p style=\"text-align: justify;\"><b>Gi\u1ea3i:<\/b><span style=\"font-weight: 400;\"> u = 5x \u21d2 u&#8217; = 5<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">y&#8217; = e^{5x} \\cdot 5 = 5e^{5x}<\/span><\/p>\n<p style=\"text-align: justify;\"><b>B\u00e0i 2:<\/b><span style=\"font-weight: 400;\"> T\u00ednh \u0111\u1ea1o h\u00e0m <\/span><span style=\"font-weight: 400;\">y = e^{-2x}<\/span><span style=\"font-weight: 400;\">.<\/span><\/p>\n<p style=\"text-align: justify;\"><b>Gi\u1ea3i:<\/b><span style=\"font-weight: 400;\"> u = -2x \u21d2 u&#8217; = -2<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">y&#8217; = e^{-2x} \\cdot (-2) = -2e^{-2x}<\/span><\/p>\n<p style=\"text-align: justify;\"><b>B\u00e0i 3:<\/b><span style=\"font-weight: 400;\"> T\u00ednh \u0111\u1ea1o h\u00e0m <\/span><span style=\"font-weight: 400;\">y = e^{x^2+1}<\/span><span style=\"font-weight: 400;\">.<\/span><\/p>\n<p style=\"text-align: justify;\"><b>Gi\u1ea3i:<\/b><span style=\"font-weight: 400;\"> u = x^2+1 \u21d2 u&#8217; = 2x<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">y&#8217; = e^{x^2+1} \\cdot 2x = 2x e^{x^2+1}<\/span><\/p>\n<p><picture><source srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/webp-express\/webp-images\/uploads\/2026\/03\/dao-ham-e-mu-u-4.jpg.webp 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/webp-express\/webp-images\/uploads\/2026\/03\/dao-ham-e-mu-u-4-300x169.jpg.webp 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/webp-express\/webp-images\/uploads\/2026\/03\/dao-ham-e-mu-u-4-768x432.jpg.webp 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" type=\"image\/webp\"><img decoding=\"async\" class=\"aligncenter size-full wp-image-530926 webpexpress-processed\" src=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/03\/dao-ham-e-mu-u-4.jpg\" alt=\"dao-ham-e-mu-u-4\" width=\"800\" height=\"450\" srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/03\/dao-ham-e-mu-u-4.jpg 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/03\/dao-ham-e-mu-u-4-300x169.jpg 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/03\/dao-ham-e-mu-u-4-768x432.jpg 768w\" sizes=\"(max-width: 800px) 100vw, 800px\"><\/picture><\/p>\n<p style=\"text-align: justify;\"><b>B\u00e0i 4:<\/b><span style=\"font-weight: 400;\"> T\u00ednh \u0111\u1ea1o h\u00e0m <\/span><span style=\"font-weight: 400;\">y = e^{\\tan x}<\/span><span style=\"font-weight: 400;\">.<\/span><\/p>\n<p style=\"text-align: justify;\"><b>Gi\u1ea3i:<\/b><span style=\"font-weight: 400;\"> u = tan x \u21d2 u&#8217; = 1 + tan^2 x<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">y&#8217; = e^{\\tan x} \\cdot (1 + tan^2 x)<\/span><\/p>\n<p style=\"text-align: justify;\"><b>B\u00e0i 5:<\/b><span style=\"font-weight: 400;\"> T\u00ednh \u0111\u1ea1o h\u00e0m <\/span><span style=\"font-weight: 400;\">y = e^{\\sin x + x^2}<\/span><span style=\"font-weight: 400;\">.<\/span><\/p>\n<p style=\"text-align: justify;\"><b>Gi\u1ea3i:<\/b><span style=\"font-weight: 400;\"> u = sin x + x^2 \u21d2 u&#8217; = cos x + 2x<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">y&#8217; = e^{\\sin x + x^2} \\cdot (cos x + 2x)<\/span><\/p>\n<h2 style=\"text-align: justify;\"><b>\u1ee8ng d\u1ee5ng th\u1ef1c t\u1ebf c\u1ee7a \u0110\u1ea1o h\u00e0m e m\u0169 u<\/b><\/h2>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">\u0110\u1ea1o h\u00e0m c\u1ee7a e m\u0169 u c\u00f3 r\u1ea5t nhi\u1ec1u \u1ee9ng d\u1ee5ng trong th\u1ef1c t\u1ebf, t\u1eeb to\u00e1n h\u1ecdc thu\u1ea7n t\u00fay \u0111\u1ebfn c\u00e1c l\u0129nh v\u1ef1c nh\u01b0 v\u1eadt l\u00fd, kinh t\u1ebf, sinh h\u1ecdc v\u00e0 k\u1ef9 thu\u1eadt.<\/span><\/p>\n<ul style=\"text-align: justify;\">\n<li style=\"font-weight: 400;\"><b>Gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh vi ph\u00e2n:<\/b><span style=\"font-weight: 400;\"> H\u00e0m <\/span><span style=\"font-weight: 400;\">e^{ax}<\/span><span style=\"font-weight: 400;\"> th\u01b0\u1eddng xu\u1ea5t hi\u1ec7n trong nghi\u1ec7m c\u1ee7a c\u00e1c ph\u01b0\u01a1ng tr\u00ecnh vi ph\u00e2n tuy\u1ebfn t\u00ednh.<\/span><\/li>\n<li style=\"font-weight: 400;\"><b>M\u00f4 h\u00ecnh t\u0103ng tr\u01b0\u1edfng:<\/b><span style=\"font-weight: 400;\"> C\u00e1c qu\u00e1 tr\u00ecnh t\u0103ng tr\u01b0\u1edfng d\u00e2n s\u1ed1, vi khu\u1ea9n, t\u00e0i s\u1ea3n t\u00e0i ch\u00ednh \u0111\u1ec1u tu\u00e2n theo m\u00f4 h\u00ecnh m\u0169, \u0111\u1ea1o h\u00e0m gi\u00fap x\u00e1c \u0111\u1ecbnh t\u1ed1c \u0111\u1ed9 t\u0103ng tr\u01b0\u1edfng t\u1ea1i m\u1ed7i th\u1eddi \u0111i\u1ec3m.<\/span><\/li>\n<li style=\"font-weight: 400;\"><b>V\u1eadt l\u00fd:<\/b><span style=\"font-weight: 400;\"> \u0110\u1ea1o h\u00e0m c\u1ee7a e m\u0169 u d\u00f9ng \u0111\u1ec3 m\u00f4 t\u1ea3 qu\u00e1 tr\u00ecnh ph\u00e2n r\u00e3 ph\u00f3ng x\u1ea1, d\u00f2ng \u0111i\u1ec7n trong m\u1ea1ch RC, t\u1ecfa nhi\u1ec7t, v.v.<\/span><\/li>\n<li style=\"font-weight: 400;\"><b>Kinh t\u1ebf:<\/b><span style=\"font-weight: 400;\"> Ph\u00e2n t\u00edch l\u00e3i k\u00e9p li\u00ean t\u1ee5c, t\u0103ng tr\u01b0\u1edfng kinh t\u1ebf, chi\u1ebft kh\u1ea5u d\u00f2ng ti\u1ec1n\u2026 \u0111\u1ec1u c\u1ea7n t\u1edbi \u0111\u1ea1o h\u00e0m c\u1ee7a e m\u0169 u.<\/span><\/li>\n<li style=\"font-weight: 400;\"><b>To\u00e1n h\u1ecdc \u1ee9ng d\u1ee5ng:<\/b><span style=\"font-weight: 400;\"> T\u1ed1i \u01b0u h\u00f3a h\u00e0m s\u1ed1, x\u00e1c su\u1ea5t th\u1ed1ng k\u00ea, gi\u1ea3i t\u00edch Fourier.<\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\"><b>V\u00ed d\u1ee5 th\u1ef1c t\u1ebf:<\/b><span style=\"font-weight: 400;\"> M\u00f4 h\u00ecnh t\u0103ng tr\u01b0\u1edfng li\u00ean t\u1ee5c: <\/span><span style=\"font-weight: 400;\">P(t) = P_0 e^{rt}<\/span><span style=\"font-weight: 400;\">, v\u1edbi <\/span><span style=\"font-weight: 400;\">r<\/span><span style=\"font-weight: 400;\"> l\u00e0 t\u1ed1c \u0111\u1ed9 t\u0103ng tr\u01b0\u1edfng. \u0110\u1ea1o h\u00e0m <\/span><span style=\"font-weight: 400;\">P'(t) = r P_0 e^{rt} = r P(t)<\/span><span style=\"font-weight: 400;\"> cho bi\u1ebft t\u1ed1c \u0111\u1ed9 thay \u0111\u1ed5i d\u00e2n s\u1ed1 t\u1ea1i th\u1eddi \u0111i\u1ec3m t.<\/span><\/p>\n<h2 style=\"text-align: justify;\"><b>C\u00e2u h\u1ecfi th\u01b0\u1eddng g\u1eb7p\u00a0<\/b><\/h2>\n<h3 style=\"text-align: justify;\"><b>\u0110\u1ea1o h\u00e0m e m\u0169 x c\u00f3 g\u00ec \u0111\u1eb7c bi\u1ec7t?<\/b><\/h3>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">V\u1edbi <\/span><span style=\"font-weight: 400;\">u = x<\/span><span style=\"font-weight: 400;\">, \u0111\u1ea1o h\u00e0m <\/span><span style=\"font-weight: 400;\">e^x<\/span><span style=\"font-weight: 400;\"> l\u1ea1i ch\u00ednh l\u00e0 <\/span><span style=\"font-weight: 400;\">e^x<\/span><span style=\"font-weight: 400;\">, do <\/span><span style=\"font-weight: 400;\">u&#8217; = 1<\/span><span style=\"font-weight: 400;\">. \u0110\u00e2y l\u00e0 t\u00ednh ch\u1ea5t \u0111\u1eb7c bi\u1ec7t ch\u1ec9 ri\u00eang h\u00e0m s\u1ed1 m\u0169 t\u1ef1 nhi\u00ean c\u00f3 \u0111\u01b0\u1ee3c.<\/span><\/p>\n<h3 style=\"text-align: justify;\"><b>L\u00e0m sao nh\u1eadn bi\u1ebft nhanh \u0111\u1ea1o h\u00e0m e m\u0169 u?<\/b><\/h3>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">Ch\u1ec9 c\u1ea7n nh\u1edb c\u00f4ng th\u1ee9c: \u0110\u1ea1o h\u00e0m b\u1eb1ng ch\u00ednh n\u00f3 nh\u00e2n v\u1edbi \u0111\u1ea1o h\u00e0m ph\u1ea7n m\u0169. T\u1ee9c l\u00e0 <\/span><span style=\"font-weight: 400;\">(e^u)&#8217; = e^u \\cdot u&#8217;<\/span><span style=\"font-weight: 400;\">.<\/span><\/p>\n<h3 style=\"text-align: justify;\"><b>\u0110\u1ea1o h\u00e0m v\u00e0 nguy\u00ean h\u00e0m kh\u00e1c nhau th\u1ebf n\u00e0o?<\/b><\/h3>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">\u0110\u1ea1o h\u00e0m c\u1ee7a e m\u0169 u l\u00e0 <\/span><span style=\"font-weight: 400;\">e^u \\cdot u&#8217;<\/span><span style=\"font-weight: 400;\"> (nh\u00e2n v\u1edbi \u0111\u1ea1o h\u00e0m ph\u1ea7n m\u0169), c\u00f2n nguy\u00ean h\u00e0m l\u00e0 <\/span><span style=\"font-weight: 400;\">e^u \/ u&#8217;<\/span><span style=\"font-weight: 400;\"> (chia cho \u0111\u1ea1o h\u00e0m ph\u1ea7n m\u0169) n\u1ebfu \u0111\u1ed5i bi\u1ebfn, ho\u1eb7c <\/span><span style=\"font-weight: 400;\">e^u + C<\/span><span style=\"font-weight: 400;\"> n\u1ebfu l\u1ea5y theo u.<\/span><\/p>\n<h3 style=\"text-align: justify;\"><b>C\u00f3 c\u00f4ng c\u1ee5 online n\u00e0o t\u00ednh \u0111\u1ea1o h\u00e0m kh\u00f4ng?<\/b><\/h3>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">C\u00f3, nhi\u1ec1u website v\u00e0 \u1ee9ng d\u1ee5ng nh\u01b0 WolframAlpha, Symbolab ho\u1eb7c c\u00e1c m\u00e1y t\u00ednh khoa h\u1ecdc c\u00f3 th\u1ec3 h\u1ed7 tr\u1ee3 t\u00ednh \u0111\u1ea1o h\u00e0m c\u1ee7a e m\u0169 u nhanh ch\u00f3ng.<\/span><\/p>\n<h3 style=\"text-align: justify;\"><b>\u0110\u1ea1o h\u00e0m c\u00f3 kh\u00f3 kh\u00f4ng?<\/b><\/h3>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">C\u00f4ng th\u1ee9c \u0111\u1ea1o h\u00e0m r\u1ea5t d\u1ec5 nh\u1edb v\u00e0 \u00e1p d\u1ee5ng, ch\u1ec9 c\u1ea7n n\u1eafm v\u1eefng quy t\u1eafc d\u00e2y chuy\u1ec1n v\u00e0 th\u00e0nh th\u1ea1o c\u00e1ch l\u1ea5y \u0111\u1ea1o h\u00e0m ph\u1ea7n m\u0169 l\u00e0 b\u1ea1n c\u00f3 th\u1ec3 gi\u1ea3i quy\u1ebft m\u1ecdi d\u1ea1ng b\u00e0i to\u00e1n li\u00ean quan.<\/span><\/p>\n<p style=\"text-align: justify;\"><b>Xem th\u00eam:<\/b><\/p>\n<ul>\n<li>\n<p class=\"elementor-heading-title elementor-size-default\"><span style=\"color: #339966;\"><strong><a style=\"color: #339966;\" href=\"https:\/\/hoanghamobile.com\/tin-tuc\/so-huu-ti-la-gi\/\">S\u1ed1 h\u1eefu t\u1ec9 l\u00e0 g\u00ec? S\u1ed1 v\u00f4 t\u1ec9 l\u00e0 g\u00ec? \u0110\u1eb7c tr\u01b0ng c\u1ee7a c\u00e1c s\u1ed1 trong To\u00e1n h\u1ecdc<\/a><\/strong><\/span><\/p>\n<\/li>\n<li>\n<p class=\"elementor-heading-title elementor-size-default\"><span style=\"color: #339966;\"><strong><a style=\"color: #339966;\" href=\"https:\/\/hoanghamobile.com\/tin-tuc\/cong-thuc-to-hop-chinh-hop\/\">B\u1ed9 c\u00f4ng th\u1ee9c Ho\u00e1n v\u1ecb \u2013 Ch\u1ec9nh h\u1ee3p \u2013 T\u1ed5 h\u1ee3p \u0111\u1ea7y \u0111\u1ee7 nh\u1ea5t trong To\u00e1n h\u1ecdc<\/a><\/strong><\/span><\/p>\n<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>T\u1eeb c\u00e1c b\u00e0i to\u00e1n t\u0103ng tr\u01b0\u1edfng, m\u00f4 h\u00ecnh v\u1eadt l\u00fd, \u0111\u1ebfn ph\u00e2n t\u00edch t\u00e0i ch\u00ednh, h\u00e0m s\u1ed1 m\u0169 xu\u1ea5t hi\u1ec7n \u1edf kh\u1eafp m\u1ecdi n\u01a1i. Trong \u0111\u00f3, \u0111\u1ea1o h\u00e0m e m\u0169 u l\u00e0 m\u1ed9t c\u00f4ng th\u1ee9c c\u01a1 b\u1ea3n nh\u01b0ng c\u1ef1c k\u1ef3 quan tr\u1ecdng, gi\u00fap ta ph\u00e2n t\u00edch t\u1ed1c \u0111\u1ed9 bi\u1ebfn thi\u00ean c\u1ee7a c\u00e1c qu\u00e1 tr\u00ecnh t\u1ef1 [&hellip;]<\/p>\n","protected":false},"author":84,"featured_media":530922,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[29235,3910,30453],"tags":[],"class_list":["post-530849","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-goc-to-mo","category-kham-pha","category-kien-thuc"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v24.2 (Yoast SEO v24.2) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u0110\u1ea1o h\u00e0m e m\u0169 u l\u00e0 g\u00ec? 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