Dérivées et intégrales

Attention à ajouter la constante

Primitive Fonction Dérivée
\[ \frac{a\blue{x}^{b + 1}}{b + 1} + c\blue{x} \] \[ a \blue{x}^b + c \] \[ b a \blue{x}^{b - 1} \]
\[ \frac{a b^{c \blue{x} + d}}{c ~ ln(b)} + e \blue{x}\] \[ ab^{c\blue{x} + d} + e \] \[ cab^{c\blue{x} +d} ln(a) \]
\[ \frac{\blue{x} log_a(\blue{x})}{ln(a)} \] \[ log_a( \blue{x} ) \] \[ \frac{1}{ \blue{x} ln(a)} \]
\[ \] \[ \blue{x}^{\blue{x}} \] \[ \blue{x}^{\blue{x}}ln(\blue{x} + 1) \]
Fonction Dérivée
\[ a \cdot f(\blue{x}) \] \[ a \cdot f'(\blue{x}) \]
\[ f(\blue{x} )\green±g(\blue{x} )\] \[ f'(\blue{x} )\green±g'(\blue{x} )\]
\[ f(\blue{x}) \cdot g(\blue{x}) \] \[ f'(\blue{x}) \cdot g(\blue{x}) + f(\blue{x}) \cdot g'(\blue{x} )\]
\[ \frac{f(\blue{x})}{g(\blue{x})} \] \[ \frac{f'(\blue{x}) g(\blue{x}) - f(\blue{x}) g'(\blue{x})}{(g(\blue{x}))^2} \]
\[ f(g(h(\blue{x}))) \] \[ f'(g(h(\blue{x}))) \cdot g'(h(\blue{x})) \cdot h'(\blue{x}) \]
\[ f^{-1}(\blue{x}) \] \[ \frac{1}{f'(f^{-1}(\blue{x}))} \]
Fonction Dérivée Fonction Dérivée
\[ sin(\blue{x}) \] \[ cos(\blue{x}) \] \[ cos(\blue{x}) \] \[ -sin(\blue{x} )\]
\[ tan(\blue{x}) \] \[ 1 + tan^2(\blue{x}) \] \[ cot(\blue{x}) \] \[ -(1 + cot^2(\blue{x})) \]
\[ sec(\blue{x}) \] \[ sec(\blue{x})tan(\blue{x}) \] \[ csc(\blue{x}) \] \[ -csc(\blue{x})cot(\blue{x}) \]
\[ sin^{-1}(\blue{x}) \] \[ \frac{1}{\sqrt{1 - x^2}} \] \[ cos^{-1}(\blue{x}) \] \[ -\frac{1}{\sqrt{1 - x^2}} \]
\[ tg^{-1}(\blue{x}) \] \[ \frac{1}{1 + \blue{x}^2} \] \[ cot^{-1}(\blue{x}) \] \[ -\frac{1}{1 + \blue{x}^2} \]
\[ sec^{-1}(\blue{x}) \] \[ \frac{1}{|\blue{x}|\sqrt{\blue{x}^2 - 1}} \] \[ csc^{-1}(\blue{x}) \] \[ -\frac{1}{|\blue{x}|\sqrt{\blue{x}^2 - 1}} \]
\[ \begin{aligned} & \int_\orange{a}^\orange{b} f(x)dx = \orange{-} \int_\orange{b}^\orange{a} f(x)dx \\[2em] & \int u' e^{u} dx = e^{u} + C \\[2em] & \int \frac{u'}{u} dx = ln(u) + C \\[2em] & \int \frac{u'}{1 + u^2} = tg^{-1} (u) + C \\[2em] & \int u'u^n dx = \frac{u^{n + 1}}{n + 1} + C \end{aligned} \]
Intégrales simples
Fonction Dérivée
\[ \frac{1}{3} \blue{u}^3 \] \[ \blue{u}^2 \blue{u}' \]
\[ \frac{1}{2} \blue{u}^2 \] \[ \blue{u} \blue{u}' \]
\[ \blue{u} \] \[ \blue{u}' \]
\[ ln(\blue{u}) \] \[ \frac{\blue{u}'}{\blue{u}} \]
\[ - \frac{1}{\blue{u}} \] \[ \frac{\blue{u}'}{\blue{u}^2} \]
\[- \frac{1}{2\blue{u}^2} \] \[ \frac{\blue{u}'}{\blue{u}^3} \]
Intégrales utiles
Fonction Dérivée
\[ \blue{x} ~ ln(\blue{x}) - x \] \[ ln(\blue{x}) \]
\[ ln(\blue{u}^n) \] \[ \frac{n}{\blue{u}} \]
\[ atg(\blue{u}) \] \[ \frac{\blue{u}'}{1 + \blue{u}^2} \]
\[ \frac{1}{a} atg(\frac{\blue{u}}{a}) \] \[ \frac{\blue{u}'}{a^2 + \blue{u}^2} \]
\[ (\blue{u} \blue{v})' - \blue{u}' \blue{v} \] \[ \blue{u} \blue{v}' \]

Taylor-Young et Maclaurin

\[ t(\blue{x}) = f(x_0) + f'(x_0)(\blue{x} - x_0) \] \[ \begin{aligned} & \text{linéarisation en } x_0 \\ & \text{fonction de la tangente en } x_0 \end{aligned}\]
\[ \begin{aligned} & T_1(\blue{x}) = f(x_0) + f'(x_0)(\blue{x} - x_0) + o(x) \\[2em] & T_2(\blue{x}) = f(x_0) + f'(x_0)(\blue{x} - x_0) + \frac{f''(x_0)}{2!}(\blue{x}-x_0) + o(x^2) \\[2em] & T_3(\blue{x}) = f(x_0) + f'(x_0)(\blue{x} - x_0) + \frac{f''(x_0)}{2!}(\blue{x}-x_0) + \frac{f'''(x_0)}{3!}(\blue{x}-x_0) +o(x^3) \end{aligned}\]
\[ T_n(\blue{x}) = \sum_{k=0}^{n} \frac{f^{(k)}(x_0)}{k!} (\blue{x} - x_0)^k + R_n(x) \] Formule générale
\[ \begin{aligned} \lim_{x \to x_0} \frac{f(x)}{g(x)} \quad = \quad \lim_{x \to x_0} \frac{F_n(x) + o(x^n)}{G_n(x) + o(x^n)} \quad \overset{x_0 = 0}{=} \quad \lim_{x \to x_0} \frac{f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + ... + \frac{f^k(0)}{k!}x^k}{g(0) + g'(0)x + \frac{g''(0)}{2!}x^2 + \frac{g'''(0)}{3!}x^3 + ... + \frac{g^k(0)}{k!}x^k} \end{aligned}\]

Méthodes

Changement de variable

Exemple 1

\[ \begin{aligned} & I = \int_{\blue{0}}^{\blue{1}} \sqrt{ 1 - \blue{x}^2 } ~~ \blue{dx} \end{aligned} \]
\[ \begin{aligned} & \blue{x} = \orange{cos(t)} \\[2em] & x' = \frac{ \blue{dx} }{ dt } = -sin(t) \Rightarrow \blue{dx} = \orange{ -sin(t)dt } \\[2em] & \blue{x} = 0 \Rightarrow \orange{t = \frac{ \pi }{ 2 }} \\[2em] & \blue{x} = 1 \Rightarrow \orange{t = 0} \end{aligned} \]
\[ \begin{aligned} & I = \int_{\orange{\frac{ \pi }{ 2 }}}^{\orange{0}} \sqrt{ 1 - \orange{cos(t)}^2 } ~~ \orange{ -sin(t)dt } = ...magie... = \frac{1}{4} \pi \end{aligned} \]

Exemple 2

\[ \begin{aligned} & I = \int_{\blue{0}}^{\blue{\frac{\pi}{4}}} tg( \blue{x} )^3 ~~ \blue{dx} \end{aligned} \]
\[ \begin{aligned} & \blue{u} = \orange{tg(x)} \\[2em] & u' = \frac{ \blue{ dx} }{ dt } = 1 + tg(x)^2 \Rightarrow \blue{dx} = \orange{ \frac{du}{1 + u^2} } \\[2em] & \blue{x} = 0 \Rightarrow \orange{ u = 0 } \\[2em] & \blue{x} = \frac{\pi}{4} \Rightarrow \orange{ u = 1 } \end{aligned} \]
\[ \begin{aligned} & I = \int_{\orange{0}}^{\orange{1}} \frac{ \orange{ u }^3 }{ 1 + \orange{ u }^2 } ~~ \orange{ du } = ...magie... = \frac{1}{2} - \frac{1}{2} ln(2) \end{aligned} \]