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Calcul de dérivées partielles

Fonctions d’une variables

Exercice 1

Calculer les dérivées partielles des fonctions suivantes :

$$ f(x,y) = x \quad g(x,y) = y \quad h(x,y) = xy $$

 

  • Première fonction

$$ \dfrac{\partial f(x,y)}{\partial x} = \dfrac{\partial}{\partial x} [x] = 1 $$
$$ \dfrac{\partial f(x,y)}{\partial y} = \dfrac{\partial}{\partial y} [x] = 0 $$

 

  • Deuxième fonction

$$ \dfrac{\partial g(x,y)}{\partial x} = \dfrac{\partial}{\partial x} [y] = 0 $$
$$ \dfrac{\partial g(x,y)}{\partial y} = \dfrac{\partial}{\partial y} [y] = 1 $$

 

  • Troisième fonction

$$ \dfrac{\partial h(x,y)}{\partial x} = \dfrac{\partial}{\partial x} [xy] = y\dfrac{\partial}{\partial x} [x] = y $$
$$ \dfrac{\partial h(x,y)}{\partial y} = \dfrac{\partial}{\partial y} [xy] = x\dfrac{\partial}{\partial y} [y] = x $$

Exercice 2

Calculer les dérivées partielles des fonctions suivantes:
$$ f(x,y) = 4x + 9y \quad g(x,y) = xy + x^3y^2 \quad h(x,y) = x^2y + xy^2 + x $$

 

  • Première fonction

$$\begin{array}{rlrl} \dfrac{\partial f(x,y)}{\partial x} & = \dfrac{\partial}{\partial x}[4x + 9y] \qquad & \dfrac{\partial f(x,y)}{\partial y} & =\dfrac{\partial }{\partial y}[4x + 9y] \\[0.1cm]& = \dfrac{\partial}{\partial x} [4x] + \dfrac{\partial}{\partial x} [9y] \qquad  & &= \dfrac{\partial}{\partial y} [4x] + \dfrac{\partial}{\partial y} [9y] \\[0.1cm]& = 4 + \underbrace{\dfrac{\partial}{\partial x} [9y]}_{=0} \qquad & & =\underbrace{\dfrac{\partial}{\partial y} [4x]}_{=0} + 9 \\[0.1cm]& = 4 \qquad & & = 9 \end{array} $$

 

  • Deuxième fonction

$$\begin{array}{rlrl} \dfrac{\partial g(x,y)}{\partial x} & = \dfrac{\partial }{\partial x}xy + x^3y^2] \qquad & \dfrac{\partial g(x,y)}{\partial y} & = \dfrac{\partial }{\partial y}xy + x^3y^2] \\[0.1cm]& = \dfrac{\partial}{\partial x} [xy] + \dfrac{\partial}{\partial x} [x^3y^2]\qquad && = \dfrac{\partial}{\partial y} [xy] + \dfrac{\partial}{\partial y} [x^3y^2] \\[0.1cm]& = y\dfrac{\partial}{\partial x} [x] + y^2\dfrac{\partial}{\partial x} [x^3]\qquad && = x\dfrac{\partial}{\partial y} [y] + x^3\dfrac{\partial}{\partial x} [y^2] \\[0.1cm]& = y + 3y^2x^2\qquad && = x + 2yx^3 \end{array} $$

 

  • Troisième fonction

$$\begin{array}{rlrl}\dfrac{\partial h(x,y)}{\partial x} & = \dfrac{\partial}{\partial x} [x^2y + xy^2 + x] \qquad & \dfrac{\partial h(x,y)}{\partial y} & = \dfrac{\partial}{\partial y} [x^2y + xy^2 + x]\\[0.1cm]& = \dfrac{\partial}{\partial x} [x^2y] + \dfrac{\partial}{\partial x} [xy^2] +\dfrac{\partial}{\partial x} [x] \qquad & & \dfrac{\partial}{\partial y} [x^2y] +\dfrac{\partial}{\partial y} [xy^2] + \dfrac{\partial}{\partial y} [x]\\[0.1cm]& = y\dfrac{\partial}{\partial x} [x^2] + y^2\dfrac{\partial}{\partial x} [x] + 1\qquad & & = x^2\dfrac{\partial}{\partial y} [y] + x\dfrac{\partial}{\partial y}[y^2] + \underbrace{\dfrac{\partial}{\partial y} [x]}_{=0}\\[0.1cm]& = 2xy + y^2 + 1 \qquad & & = x^2 + 2xy\end{array}$$

 

Exercice 3

Calculer les dérivées partielles des fonctions suivantes:
$$ f(x,y) = \sqrt{x^2 + y^2} \qquad g(x,y) = \dfrac{1}{x+y} \qquad h(x,y) = \dfrac{x^2 + y^3}{y – x + 2} $$

 

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Exercice 4

Calculer les dérivées partielles des fonctions suivantes:
$$ f(x,y) = e^{4x^3 + 3y} \qquad g(x,y) = (y-4)e^x \qquad h(x,y) = (x^3 + y^4)e^x $$

 

Première fonction

$$
\begin{array}{rlrl}
\dfrac{\partial f(x,y)}{\partial x} & = \dfrac{\partial}{\partial x}[e^{4x^3 + 3y}] \qquad & \dfrac{\partial f(x,y)}{\partial x} & = \dfrac{\partial}{\partial x}[e^{4x^3 + 3y}]\\[0.1cm]
& = \dfrac{\partial}{\partial x}[4x^3 + 3y] e^{4x^3 + 3y} \qquad & & = \dfrac{\partial}{\partial y}[4x^3 + 3y] e^{4x^3 + 3y}\\[0.1cm]
& = (\dfrac{\partial}{\partial x}[4x^3] + \underbrace{\dfrac{\partial}{\partial x}[3y]}_{=0}) e^{4x^3 + 3y} \qquad & & = (\underbrace{\dfrac{\partial}{\partial y}[4x^3]}_{=0} + \dfrac{\partial}{\partial y}[3y]) e^{4x^3 + 3y}\\[0.1cm]
& = 12x^2 e^{4x^3 + 3y} \qquad & & = 3 e^{4x^3 + 3y}
\end{array}
$$

 

Deuxième fonction

$$
\begin{array}{rlrl}
\dfrac{\partial g(x,y)}{\partial x} & = \dfrac{\partial}{\partial x}[(y-4)e^x] \qquad & \dfrac{\partial g(x,y)}{\partial y} & = \dfrac{\partial}{\partial y}[(y-4)e^x] \\[0.1cm]
& = (y-4) \dfrac{\partial}{\partial x}[e^x] \qquad & & = \dfrac{\partial}{\partial y}[(y-4)]e^x \\[0.1cm]
& = (y-4) e^x \qquad & & = e^x
\end{array}
$$

 

Troisième fonction

$$
\begin{array}{rlrl}
\dfrac{\partial h(x,y)}{\partial x} & = \dfrac{\partial}{\partial x}[(x^3 + y^4)e^x] \qquad & \dfrac{\partial h(x,y)}{\partial y} & = \dfrac{\partial}{\partial y}[(x^3 + y^4)e^x] \\[0.1cm]
& = \dfrac{\partial}{\partial x}[(x^3 + y^4)]e^x + \dfrac{\partial}{\partial x}[e^x](x^3 + y^4) \qquad & & = \dfrac{\partial}{\partial y}[(x^3 + y^4)]e^x \\[0.1cm]
& = 3x^2 e^x + (x^3 + y^4) e^x \qquad & & = 4y^3 e^x
\end{array}
$$

Exercice 4

Calculer les dérivées partielles des fonctions suivantes:
$$ f(x,y) = \ln(x+y) \qquad g(x,y) = \dfrac{1}{2} \ln{5x^2 – 2y} \qquad h(x,y) = (x+y^2)\ln{x+y} $$

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