{"id":85215,"date":"2020-12-24T14:46:06","date_gmt":"2020-12-24T13:46:06","guid":{"rendered":"https:\/\/groupe-reussite.fr\/?page_id=85215"},"modified":"2023-09-15T13:09:57","modified_gmt":"2023-09-15T12:09:57","slug":"cours-en-ligne-derivation-premiere","status":"publish","type":"page","link":"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-derivation-premiere\/","title":{"rendered":"D\u00e9rivation en premi\u00e8re"},"content":{"rendered":"
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\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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Cours en ligne Maths en Premi\u00e8re<\/a><\/p>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t

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Chapitres de maths en 1\u00e8re<\/p>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t

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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tD\u00e9rivation<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tVariations et courbes repr\u00e9sentatives de fonctions<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tFonction exponentielle<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tCalcul vectoriel et produit scalaire<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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D\u00e9rivation en 1\u00e8re<\/h1>\n<\/div>\n

R\u00e9sum\u00e9 de cours<\/a> Exercices et corrig\u00e9s<\/a><\/p>\n

Cours en ligne de Maths en Premi\u00e8re<\/a><\/p>\n

Ce cours en ligne de maths en premi\u00e8re<\/a> pr\u00e9sente est utile pour les \u00e9l\u00e8ves souhaitant s’entra\u00eener \u00e0 la d\u00e9rivation en premi\u00e8re<\/strong> : nombre d\u00e9riv\u00e9<\/strong>, fonction d\u00e9riv\u00e9e<\/strong>, formule d\u00e9riv\u00e9e,<\/strong> etc.. D’autres chapitres et des profs de maths en ligne<\/a> de premi\u00e8re sont \u00e9galement disponibles sur notre site comme, le second degr\u00e9<\/a>, les suites num\u00e9riques<\/a> ou encore les s\u00e9ries arithm\u00e9tiques et g\u00e9om\u00e9triques<\/a>.<\/p>\n

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Nombre d\u00e9riv\u00e9 : d\u00e9finition et exemples fondamentaux<\/h2>\n

D\u00e9finition d’un nombre d\u00e9riv\u00e9<\/h3>\n

Soit \"f\" une fonction d\u00e9finie sur un intervalle \"I\", et soit \"a \in I.\" On d\u00e9finit le nombre d\u00e9riv\u00e9<\/strong> de \"f\" en \"a\" le nombre not\u00e9 \"f'(a)\" tel que :<\/p>\n

\"f'(a) = \lim\limits_{h \rightarrow 0}\dfrac{f(a+h)-f(a)}{h}\"<\/p>\n

Nombre d\u00e9riv\u00e9 : exemples fondamentaux<\/h3>\n

Exemple 1 : la fonction carr\u00e9<\/strong><\/p>\n

Posons \"f(x) = x^2\", et calculons le taux d\u2019accroissement de \"f\" entre \"a\" et \"a + h\", o\u00f9 a est un nombre quelconque et \"h > 0\" :<\/p>\n

\"τa(h) =\dfrac{f(a +h)− f(a)}{h}\"<\/p>\n

\"= \dfrac{(a +h)^{2}−a^{2}}{h}\"<\/p>\n

\"= \dfrac{a^{2} +2ah +h^{2} −a^{2}}{h}\"<\/p>\n

\"= \dfrac{h(2a +h)}{h}\"<\/p>\n

\"= 2a +h\"<\/p>\n

Ainsi, \"f'(a)\"<\/p>\n

\"=\lim\limits_{h \rightarrow 0} 2a+h\"<\/p>\n

\"= 2a +0\"<\/p>\n

\"= 2a.\"<\/p>\n

Quelle que soit la valeur de \"a\", le nombre d\u00e9riv\u00e9<\/strong> de \"x^{2}\" en a est \"f'(a) = 2a\"<\/p>\n

Exemple 2 : la fonction cube<\/strong><\/p>\n

Posons \"f (x) = x^{3}\", et calculons le taux d\u2019accroissement de \"f\" entre \"a\" et \"a + h\", o\u00f9 \"a\" est un nombre quelconque et \"h > 0\" :<\/p>\n

\"τa(h) =\dfrac{f(a +h)− f(a)}{h}\"<\/p>\n

\"= \dfrac{(a+h)^{3} − a^{3}}{h}\"<\/p>\n

\"= \dfrac{a^{3} +3a^{2}h +3ah^{2} +h^{3} −a^{3}}{h}\"<\/p>\n

\"= \dfrac{h(3a^{2} +3ah +h^{2}}{h}\"<\/p>\n

\"= 3a^{2} +3ah +h^{2}+{h}\"<\/p>\n

Ainsi, \"f'(a)\"<\/p>\n

\"= \lim\limits_{h \rightarrow 0} 3a^{2}+ 3ah+ h^{2} +h\"<\/p>\n

\"= 3a^{2}\"<\/p>\n

Quelle que soit la valeur de \"a\", le nombre d\u00e9riv\u00e9<\/strong> de \"x^{3}\" en \"a\" est \"f'(a)=3a^{2}.\"<\/p>\n

Exemple 3 : la fonction racine carr\u00e9e<\/strong><\/p>\n

Posons \"f (x) = \sqrt(x)\"<\/p>\n

\"τa(h) =\dfrac{f(a +h)− f(a)}{h}\"<\/p>\n

\"=\dfrac{\sqrt(a+h)-\sqrt(a)}{h}\"<\/p>\n

\"=\dfrac{\sqrt(a+h)-\sqrt(a)}{h}\" x \"\dfrac{\sqrt(a+h)+\sqrt(a)}{\sqrt(a+h)+\sqrt(a)}\"<\/p>\n

\"=\dfrac{a+h-a}{h\times(\sqrt(a+h)+\sqrt(a))}\"<\/p>\n

\"=\dfrac{h}{h\times(\sqrt(a+h)+\sqrt(a))}\"<\/p>\n

\"=\dfrac{1}{\sqrt(a+h)+\sqrt(a)}\"<\/p>\n

Ainsi \"f'(a)\"<\/p>\n

\"= \lim\limits_{h \rightarrow 0} \dfrac{1}{\sqrt(a+h)+\sqrt(a)}\"<\/p>\n

\"=\dfrac{1}{2\sqrt(a)}\"<\/p>\n

Quelle que soit la valeur de \"a\", le nombre d\u00e9riv\u00e9<\/strong> de \"\sqrt{x}\" en \"a\" est :<\/p>\n

\"f'(a)=\dfrac{1}{2\sqrt(a)}\"<\/p>\n

Exemple 4 : la fonction inverse<\/strong><\/p>\n

Posons \"f(x)=\dfrac{1}{x}\"<\/p>\n

\"τa(h) =\dfrac{f(a +h)− f(a)}{h}\"<\/p>\n

\"=\dfrac{\dfrac{1}{a+h}-\dfrac{1}{a}}{h}\"<\/p>\n

\"=\dfrac{\dfrac{a-(a+h)}{a(a+h)}}{h}\"<\/p>\n

\"=-\dfrac{h}{a\times(a+h)}\times\dfrac{1}{h}\"<\/p>\n

\"=-\dfrac{1}{a(a+h)}\"<\/p>\n

Ainsi \"f'(a)\"<\/p>\n

\"= \lim\limits_{h \rightarrow 0} -\dfrac{1}{a(a+h)}\"<\/p>\n

\"=-\dfrac{1}{a^{2}}\"<\/p>\n

Quelle que soit la valeur de \"a\", le nombre d\u00e9riv\u00e9<\/strong> de \"\dfrac{1}{x}\" en \"a\" est \"f'(a)=-\dfrac{1}{a^{2}}\"<\/p>\n

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UN PROF DE MATHS POUR EXCELLER<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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La pratique et la compr\u00e9hension
cl\u00e9s de la r\u00e9ussite<\/b><\/h4>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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Cours de maths en ligne ou \u00e0 domicile<\/h4>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tProfesseur particulier de maths<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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Avis Google France <\/b>\u2605\u2605\u2605\u2605\u2605<\/b><\/span> 4,9 sur 5<\/b><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t

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 <\/p>\n

Equation de la tangente<\/h2>\n

D\u00e9finition d’une tangente<\/h3>\n

Soit \"C\" la courbe repr\u00e9sentative d\u2019une fonction \"f\" d\u00e9finie sur un intervalle \"I\". Soit \"a \in I\". On appelle tangente \u00e0 \"C\" au point d\u2019abscisse a la droite de coefficient directeur \"f'(a)\" passant par le point de coordonn\u00e9es \"(a;f(a)).\"<\/p>\n

Equation d’une tangente<\/h3>\n

Soit \"C\" la courbe repr\u00e9sentative d\u2019une fonction \"f\" d\u00e9finie sur un intervalle \"I\". Soit \"a \in I\". L\u2019\u00e9quation r\u00e9duite de la tangente \u00e0 \"C\" au point d\u2019abscisse a est : \"y=f'(a)(x-a)+f(a)\"<\/p>\n

Fonction d\u00e9riv\u00e9e<\/h2>\n

D\u00e9finition d’une fonction d\u00e9riv\u00e9e<\/h3>\n

Soit \"f\" une fonction d\u00e9finie sur un intervalle \"I\". On d\u00e9finit la fonction d\u00e9riv\u00e9e<\/strong> de \"f\" comme \u00e9tant la fonction : \"f': x → f'(x)\" o\u00f9 \"f'(x)\" est le nombre d\u00e9riv\u00e9 de \"f\" en \"x\". Si \"f'(x)\" est d\u00e9finie sur un intervalle \"J\" inclus dans I alors on dit que \"f\" est d\u00e9rivable sur \"J\". Attention \u00e0 ne pas confondre fonction d\u00e9riv\u00e9e et fonction exponentielle<\/a> !<\/p>\n

D\u00e9riv\u00e9es de r\u00e9f\u00e9rence<\/h3>\n

D\u2019apr\u00e8s les exemples 1, 2, 3 et 4, on peut \u00e9crire :<\/p>\n

1. Si \"f(x) = x^{2}\" alors \"f'(x) = 2x\" sur \"\mathbb{R}\".<\/p>\n

2. Si \"f(x) = x^{3}\" alors \"f'(x) = 3x^{2}\" sur \"\mathbb{R}\".<\/p>\n

3. Si \"f(x) = \sqrt(x)\" alors \"f'(x) = \dfrac{1}{2\sqrt(x)}\" sur \"]0;+\infty]\" (la fonction n\u2019est pas d\u00e9rivable<\/strong> en 0).<\/p>\n

4. Si \"f (x) = \dfrac{1}{x}\" alors \"f'(x) = \dfrac{-1}{2\sqrt(x)}\" sur \"]−\infty; 0[\" et sur \"]0;+\infty[\".<\/p>\n

5. Si \"f(x) = x^{n}, n \in I\", alors \"f'(x) = nxn^{-1}\"<\/p>\n

D\u00e9riv\u00e9e d’une fonction compos\u00e9e<\/h3>\n

Soit la fonction \"x → g(ax +b)\", o\u00f9 \"a\" et \"b\" sont deux nombres r\u00e9els. Alors, sa fonction d\u00e9riv\u00e9e<\/strong> est : \"x → ag'(ax +b).\"<\/p>\n

Exemple<\/strong><\/p>\n

\"f (x) = \sqrt(−5x +20)\", d\u00e9finie sur \"]−\infty; 4]\".<\/p>\n

Ici, \"g(x) = \sqrt(x)\" et \"f (x) = g(−5x+20)\".<\/p>\n

\"g'(x) = \dfrac{1}{2\sqrt(x)}\" donc \"f'(x) =-5g'(−5x +20)\" soit :<\/p>\n

\"f'(x) =\dfrac{-5}{2\sqrt(-5x+20)}\", d\u00e9finie sur \"]−\infty; 4[.\"<\/p>\n

1. \"(ku)' = ku'\"<\/p>\n

2. \"(u + v)' = u' + v'\"<\/p>\n

3. \"(u − v)' = u' − v'\"<\/p>\n

4. \"(uv)' = u'v +uv'\"<\/p>\n

5. \"\dfrac{u}{v}= \dfrac {u'v-uv'}{v^2}\"<\/p>\n

Exemple (produit d’un nombre et d’une fonction)<\/strong><\/p>\n

\"f (x) = 3x^{2}.\"<\/p>\n

On pose alors :<\/p>\n

\"u(x) = x^{2}\" et \"k = 3.\" Comme \"u'(x) = 2x\", on a :<\/p>\n

\"f'(x) = 3×2x = 6x.\"<\/p>\n

Exemple (diff\u00e9rence)<\/strong><\/p>\n

\"f(x) = 8x^{5} −5x^{2}.\"<\/p>\n

La d\u00e9riv\u00e9e<\/strong> \"f'(x)\" est<\/p>\n

a. \"= 40x^{4} −20x.\"<\/p>\n

b. \"= 40x^{4} −10x.\"<\/p>\n

c. \"= 40x^{2} −10x.\"<\/p>\n

Variation d’une fonction d\u00e9riv\u00e9e<\/h2>\n

Soit \"f\" une fonction d\u00e9rivable<\/strong> sur un intervalle \"I\".<\/p>\n

\"\bullet f\" est strictement croissante sur \"I \Leftrightarrow f'(x) > 0\" pour tout \"x\" de \"I.\"<\/p>\n

\"\bullet f\" est strictement d\u00e9croissante sur \"I \Leftrightarrow f'(x) < 0\" pour tout \"x\" de \"I\".<\/p>\n

Cons\u00e9quence : pour \u00e9tudier les variations d\u2019une fonction, il suffit d\u2019\u00e9tudier le signe de sa d\u00e9riv\u00e9e<\/strong>.<\/p>\n

Exemple<\/strong><\/p>\n

Soit \"f (x) = 3x^{3} −5x^{2} +4x −1.\"<\/p>\n

Sa d\u00e9riv\u00e9e<\/strong> est : \"f'(x) = 9x^{2} − 10x + 4\". C\u2019est un polyn\u00f4me de second degr\u00e9<\/a>, dont le discriminant est :<\/p>\n

\"\Delta = (−10)^{2} −4×9×4\"<\/p>\n

\"= −44 < 0.\"<\/p>\n

Ainsi, \"f'(x)\" est du signe du coefficient de \"x^{2}\", c\u2019est-\u00e0-dire ici positif.<\/p>\n

\"\forall x \in \mathbb{R}, f'(x) > 0\" donc \"f\" est strictement croissante sur \"\mathbb{R}\".<\/p>\n

 <\/p>\n\t\t

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COURS DE MATHS<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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Nous avons recrut\u00e9 pour vous les meilleurs professeurs particuliers de maths<\/b><\/h4>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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S'EXERCER ET APPRENDRE<\/h4>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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Avis Google France <\/b>\u2605\u2605\u2605\u2605\u2605<\/b><\/span> 4,8 sur 5<\/b><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t

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 <\/p>\n

Extremum local d’une fonction d\u00e9riv\u00e9e<\/h2>\n

Soit \"f\" une fonction d\u00e9rivable<\/strong> sur un intervalle \"I\". Soit \"a \in I.\"<\/p>\n

On dit que \"f\" admet un extremum local en \"a\" si \"f'(a) = 0\" et si, pour \"h \ne 0\", \"f'(a − h)\" et \"f'(a +h)\" n\u2019ont pas le m\u00eame signe.<\/p>\n

Cet extremum local peut \u00eatre :<\/p>\n

\"\bullet\" un minimum si : \"\begin{cases} f'(x)<0 \qquad pour \qquad x < a\\ f'(x)>0 \qquad pour \qquad x > a \end{cases}\"<\/p>\n

\"\bullet\" un maximum si : \"\begin{cases} f'(x)>0 \qquad pour \qquad x < a\\ f'(x)<0 \qquad pour \qquad x > a \end{cases}\"<\/p>\n

Le cours complet sur la d\u00e9rivation en 1\u00e8re<\/strong> est disponible sur notre application mobile PrepApp.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

D\u00e9rivation en 1\u00e8re R\u00e9sum\u00e9 de cours Exercices et corrig\u00e9s Cours en ligne de Maths en Premi\u00e8re Ce cours en ligne de maths en premi\u00e8re pr\u00e9sente est utile pour les \u00e9l\u00e8ves souhaitant s’entra\u00eener \u00e0 la d\u00e9rivation en premi\u00e8re : nombre d\u00e9riv\u00e9, fonction d\u00e9riv\u00e9e, formule d\u00e9riv\u00e9e, etc.. D’autres chapitres et des profs de maths en ligne de […]<\/p>\n","protected":false},"author":6,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"ocean_post_layout":"full-screen","ocean_both_sidebars_style":"","ocean_both_sidebars_content_width":0,"ocean_both_sidebars_sidebars_width":0,"ocean_sidebar":"ressource","ocean_second_sidebar":"0","ocean_disable_margins":"enable","ocean_add_body_class":"","ocean_shortcode_before_top_bar":"","ocean_shortcode_after_top_bar":"","ocean_shortcode_before_header":"","ocean_shortcode_after_header":"","ocean_has_shortcode":"","ocean_shortcode_after_title":"","ocean_shortcode_before_footer_widgets":"","ocean_shortcode_after_footer_widgets":"","ocean_shortcode_before_footer_bottom":"","ocean_shortcode_after_footer_bottom":"","ocean_display_top_bar":"default","ocean_display_header":"default","ocean_header_style":"","ocean_center_header_left_menu":"0","ocean_custom_header_template":"0","ocean_custom_logo":0,"ocean_custom_retina_logo":0,"ocean_custom_logo_max_width":0,"ocean_custom_logo_tablet_max_width":0,"ocean_custom_logo_mobile_max_width":0,"ocean_custom_logo_max_height":0,"ocean_custom_logo_tablet_max_height":0,"ocean_custom_logo_mobile_max_height":0,"ocean_header_custom_menu":"0","ocean_menu_typo_font_family":"0","ocean_menu_typo_font_subset":"","ocean_menu_typo_font_size":0,"ocean_menu_typo_font_size_tablet":0,"ocean_menu_typo_font_size_mobile":0,"ocean_menu_typo_font_size_unit":"px","ocean_menu_typo_font_weight":"","ocean_menu_typo_font_weight_tablet":"","ocean_menu_typo_font_weight_mobile":"","ocean_menu_typo_transform":"","ocean_menu_typo_transform_tablet":"","ocean_menu_typo_transform_mobile":"","ocean_menu_typo_line_height":0,"ocean_menu_typo_line_height_tablet":0,"ocean_menu_typo_line_height_mobile":0,"ocean_menu_typo_line_height_unit":"","ocean_menu_typo_spacing":0,"ocean_menu_typo_spacing_tablet":0,"ocean_menu_typo_spacing_mobile":0,"ocean_menu_typo_spacing_unit":"","ocean_menu_link_color":"","ocean_menu_link_color_hover":"","ocean_menu_link_color_active":"","ocean_menu_link_background":"","ocean_menu_link_hover_background":"","ocean_menu_link_active_background":"","ocean_menu_social_links_bg":"","ocean_menu_social_hover_links_bg":"","ocean_menu_social_links_color":"","ocean_menu_social_hover_links_color":"","ocean_disable_title":"on","ocean_disable_heading":"on","ocean_post_title":"","ocean_post_subheading":"","ocean_post_title_style":"","ocean_post_title_background_color":"","ocean_post_title_background":0,"ocean_post_title_bg_image_position":"","ocean_post_title_bg_image_attachment":"","ocean_post_title_bg_image_repeat":"","ocean_post_title_bg_image_size":"","ocean_post_title_height":0,"ocean_post_title_bg_overlay":0.5,"ocean_post_title_bg_overlay_color":"","ocean_disable_breadcrumbs":"on","ocean_breadcrumbs_color":"","ocean_breadcrumbs_separator_color":"","ocean_breadcrumbs_links_color":"","ocean_breadcrumbs_links_hover_color":"","ocean_display_footer_widgets":"default","ocean_display_footer_bottom":"default","ocean_custom_footer_template":"0","footnotes":""},"class_list":["post-85215","page","type-page","status-publish","entry"],"yoast_head":"\nD\u00e9rivation 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