Dérivées et intégrales
Formules
Dérivées
Attention à ajouter la constante d'intégration \( C \)
\(a, b, c, d, e \) sont des paramètres (même \(e\))
| 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 |
|
\[ sin(\blue{x}) \]
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\[ cos(\blue{x}) \]
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\[ cos(\blue{x}) \]
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\[ -sin(\blue{x} )\]
|
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\[ tan(\blue{x}) \]
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\[ 1 + tan^2(\blue{x}) \]
\[ \frac{1}{cos^2(\blue{x})} \]
|
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\[ cot(\blue{x}) \]
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\[ -(1 + cot^2(\blue{x})) \]
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\[ sec(\blue{x}) \]
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\[ sec(\blue{x})tan(\blue{x}) \]
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\[ csc(\blue{x}) \]
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\[ -csc(\blue{x})cot(\blue{x}) \]
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\[ sin^{-1}(\blue{x}) \]
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\[ \frac{1}{\sqrt{1 - x^2}} \]
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\[ cos^{-1}(\blue{x}) \]
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\[ -\frac{1}{\sqrt{1 - x^2}} \]
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\[ tg^{-1}(\blue{x}) \]
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\[ \frac{1}{1 + \blue{x}^2} \]
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\[ cot^{-1}(\blue{x}) \]
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\[ -\frac{1}{1 + \blue{x}^2} \]
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\[ sec^{-1}(\blue{x}) \]
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\[ \frac{1}{|\blue{x}|\sqrt{\blue{x}^2 - 1}} \]
|
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\[ csc^{-1}(\blue{x}) \]
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\[ -\frac{1}{|\blue{x}|\sqrt{\blue{x}^2 - 1}} \]
|
Intégrales
\[ \begin{aligned}
& \int_\orange{a}^\orange{b} f(x)dx = \orange{-} \int_\orange{b}^\orange{a} f(x)dx \\[2em]
& \int u' e^{u} dx = e^{u} + C \\[2em]
& \int \frac{u'}{u} dx = ln(u) + C \\[2em]
& \int \frac{u'}{1 + u^2} = tg^{-1} (u) + C \\[2em]
& \int u'u^n dx = \frac{u^{n + 1}}{n + 1} + C
\end{aligned} \]
| 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 |
\[ \begin{aligned}
\lim_{x \to x_0} \frac{f(x)}{g(x)} \quad = \quad \lim_{x \to x_0} \frac{F_n(x) + o(x^n)}{G_n(x) +
o(x^n)}
\quad \overset{x_0 = 0}{=}
\quad \lim_{x \to x_0} \frac{f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + ... +
\frac{f^k(0)}{k!}x^k}{g(0) + g'(0)x + \frac{g''(0)}{2!}x^2 + \frac{g'''(0)}{3!}x^3 + ... +
\frac{g^k(0)}{k!}x^k}
\end{aligned}\]
Techniques
Intégration par parties
\[ (fg)' = f'g + fg'
\quad\Longleftrightarrow\quad\quad
\int fg' ~~=~~ fg ~~-~~ \int fg' \]
\[ \begin{aligned}
& \int ln(x) ~dx
\quad
\begin{array}{l}
& x && \overset{'}{\rightarrow} && 1 \\[1em]
& ln(x) && \overset{'}{\rightarrow} && \frac{1}{x}
\end{array}
\\[3em]
& \int 1 ~ ln(x) ~ dx \quad=\quad
x ~ ln(x) ~~-~~ \int x \frac{1}{x} ~ dx \\[2em]
& \int ~ ln(x) ~ dx \quad=\quad
x ~ ln(x) ~~-~~ \int 1 ~ dx \\[2em]
& \int ln(x) ~ dx \quad=\quad
x ~ ln(x) ~~-~~ x \\[2em]
\end{aligned} \]
Changement de variable
Rappel
\[ f(x) = y \quad\Rightarrow\quad f'(x) = \frac{dy}{dx} \]
\[ \begin{aligned}
& I = \int_{\green{0}}^{\green{1}} \sqrt{ 1 - \green{y}^2 } ~~ \green{dy}
\\[2em]
& \green{y} = cos(\orange{x})
\quad\Rightarrow\quad
cos'(x) = \frac{\green{dy}}{\orange{dx}} = -sin(\orange{x})
\quad\Rightarrow\quad
\green{dy} =-sin(\orange{x}) \orange{dx}
\\[1em]
&
\green{y} = \green{0} \quad\Longleftrightarrow\quad \orange{x} = \orange{\frac{ \pi }{ 2 }} \\[1em]
&
\green{y} = \green{1} \quad\Longleftrightarrow\quad \orange{x} = \orange{0}
\\[2em]
&
I = \int_{\orange{\pi / 2}}^{\orange{0}}
\sqrt{1 - cos(\orange{x})^2 } ~~
(-sin(\orange{x})) \orange{dx}
\gray{
\quad= - \int_{\pi / 2}^{0} \sqrt{sin(x)^2} ~ sin(x) ~ dx =
\int_{0}^{\pi / 2} sin(x)^2 ~ dx
}
\\[2em]
& = \underline{\underline{\frac{1}{4} \pi}}
\end{aligned} \]
\[ \begin{aligned}
& I = \int_{\orange{0}}^{\orange{\frac{\pi}{4}}} tg(\orange{x})^3 ~~ \orange{dx}
\\[2em]
& \green{y} = tg(\orange{x})
\quad\Rightarrow\quad
tg'(\orange{x}) = \frac{\green{dy}}{\orange{dx}} =
1 + tg(\orange{x})^2 = 1 + \green{y}^2
\quad\Rightarrow\quad
\orange{dx} = \frac{1}{1 + \green{y}^2} ~ \green{dy}
\\[1em]
&
\green{y} = \green{0} \quad\Longleftrightarrow\quad \orange{x} = \orange{0}
\gray{\quad\Longleftrightarrow y = tg(0)} \\[1em]
&
\green{y} = \green{1} \quad\Longleftrightarrow\quad \orange{x} = \orange{\frac{\pi}{4}}
\gray{\quad\Longleftrightarrow y = tg(\frac{\pi}{4})}
\\[2em]
& I = \int_{\green{0}}^{\green{1}}
\green{y}^3 \frac{1}{1 + \green{y}^2} ~ \green{dy}
\gray{
\quad=\quad \int_{0}^{1} \frac{y^3}{1 +y^2 } ~~ dy
\quad=\quad \int_{0}^{1} y - \frac{y}{1 +y^2 } ~~ dy
\quad=\quad \int_{0}^{1} y ~ dy - \int_{0}^{1}\frac{y}{1 +y^2 } ~~ dy
\quad=\quad \left[ \frac{1}{2} \right]_0^1 - \left[ \frac{1}{2} ln(1 + y^2)\right]_0^1
}
\\[2em]
& = \underline{\underline{ \frac{1}{2} - \frac{1}{2} ln(2) }}
\end{aligned} \]