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Calculatrice de dérivées partielles
Calculez les dérivées partielles étape par étape
Cette calculatrice en ligne calculera la dérivée partielle de la fonction, avec les étapes détaillées. Vous pouvez spécifier n'importe quel ordre d'intégration.
Solution
Your input: find $$$\frac{\partial^{2}}{\partial y^{2}}\left(e^{x y}\right)$$$
First, find $$$\frac{\partial}{\partial y}\left(e^{x y}\right)$$$
Write the function $$$e^{x y}$$$ as a composition of the two functions $$$u=g=x y$$$ and $$$f\left(u\right)=e^{u}$$$.
Apply the chain rule $$$\frac{\partial}{\partial y} \left(f\left(g\right) \right)=\frac{\partial}{\partial u} \left(f\left(u\right) \right) \cdot \frac{\partial}{\partial y} \left(g \right)$$$:
$${\color{red}{\frac{\partial}{\partial y}\left(e^{x y}\right)}}={\color{red}{\frac{\partial}{\partial u}\left(e^{u}\right) \frac{\partial}{\partial y}\left(x y\right)}}$$The derivative of an exponential is $$$\frac{\partial}{\partial u} \left(e^{u} \right)=e^{u}$$$:
$${\color{red}{\frac{\partial}{\partial u}\left(e^{u}\right)}} \frac{\partial}{\partial y}\left(x y\right)={\color{red}{e^{u}}} \frac{\partial}{\partial y}\left(x y\right)$$Return to the old variable:
$$e^{{\color{red}{u}}} \frac{\partial}{\partial y}\left(x y\right)=e^{{\color{red}{x y}}} \frac{\partial}{\partial y}\left(x y\right)$$Apply the constant multiple rule $$$\frac{\partial}{\partial y} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial y} \left(f \right)$$$ with $$$c=x$$$ and $$$f=y$$$:
$$e^{x y} {\color{red}{\frac{\partial}{\partial y}\left(x y\right)}}=e^{x y} {\color{red}{x \frac{\partial}{\partial y}\left(y\right)}}$$Apply the power rule $$$\frac{\partial}{\partial y} \left(y^{n} \right)=n\cdot y^{-1+n}$$$ with $$$n=1$$$, in other words $$$\frac{\partial}{\partial y} \left(y \right)=1$$$:
$$x e^{x y} {\color{red}{\frac{\partial}{\partial y}\left(y\right)}}=x e^{x y} {\color{red}{1}}$$Thus, $$$\frac{\partial}{\partial y}\left(e^{x y}\right)=x e^{x y}$$$
Next, $$$\frac{\partial^{2}}{\partial y^{2}}\left(e^{x y}\right)=\frac{\partial}{\partial y} \left(\frac{\partial}{\partial y}\left(e^{x y}\right) \right)=\frac{\partial}{\partial y}\left(x e^{x y}\right)$$$
Apply the constant multiple rule $$$\frac{\partial}{\partial y} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial y} \left(f \right)$$$ with $$$c=x$$$ and $$$f=e^{x y}$$$:
$${\color{red}{\frac{\partial}{\partial y}\left(x e^{x y}\right)}}={\color{red}{x \frac{\partial}{\partial y}\left(e^{x y}\right)}}$$Write the function $$$e^{x y}$$$ as a composition of the two functions $$$u=g=x y$$$ and $$$f\left(u\right)=e^{u}$$$.
Apply the chain rule $$$\frac{\partial}{\partial y} \left(f\left(g\right) \right)=\frac{\partial}{\partial u} \left(f\left(u\right) \right) \cdot \frac{\partial}{\partial y} \left(g \right)$$$:
$$x {\color{red}{\frac{\partial}{\partial y}\left(e^{x y}\right)}}=x {\color{red}{\frac{\partial}{\partial u}\left(e^{u}\right) \frac{\partial}{\partial y}\left(x y\right)}}$$The derivative of an exponential is $$$\frac{\partial}{\partial u} \left(e^{u} \right)=e^{u}$$$:
$$x {\color{red}{\frac{\partial}{\partial u}\left(e^{u}\right)}} \frac{\partial}{\partial y}\left(x y\right)=x {\color{red}{e^{u}}} \frac{\partial}{\partial y}\left(x y\right)$$Return to the old variable:
$$x e^{{\color{red}{u}}} \frac{\partial}{\partial y}\left(x y\right)=x e^{{\color{red}{x y}}} \frac{\partial}{\partial y}\left(x y\right)$$Apply the constant multiple rule $$$\frac{\partial}{\partial y} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial y} \left(f \right)$$$ with $$$c=x$$$ and $$$f=y$$$:
$$x e^{x y} {\color{red}{\frac{\partial}{\partial y}\left(x y\right)}}=x e^{x y} {\color{red}{x \frac{\partial}{\partial y}\left(y\right)}}$$Apply the power rule $$$\frac{\partial}{\partial y} \left(y^{n} \right)=n\cdot y^{-1+n}$$$ with $$$n=1$$$, in other words $$$\frac{\partial}{\partial y} \left(y \right)=1$$$:
$$x^{2} e^{x y} {\color{red}{\frac{\partial}{\partial y}\left(y\right)}}=x^{2} e^{x y} {\color{red}{1}}$$Thus, $$$\frac{\partial}{\partial y}\left(x e^{x y}\right)=x^{2} e^{x y}$$$
Therefore, $$$\frac{\partial^{2}}{\partial y^{2}}\left(e^{x y}\right)=x^{2} e^{x y}$$$
Answer: $$$\frac{\partial^{2}}{\partial y^{2}}\left(e^{x y}\right)=x^{2} e^{x y}$$$