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Rechner für partielle Ableitungen
Partielle Ableitungen Schritt für Schritt berechnen
Dieser Online-Rechner berechnet die partielle Ableitung der Funktion und zeigt die Rechenschritte an. Sie können jede Ableitungsordnung angeben.
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: $$$\frac{\partial^{2}}{\partial y^{2}}\left(e^{x y}\right)=x^{2} e^{x y}$$$