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편도함수 계산기
편도함수를 단계별로 계산
이 온라인 계산기는 풀이 과정을 보여 주면서 함수의 편미분을 계산합니다. 적분의 순서는 임의로 지정할 수 있습니다.
Solution
Your input: find $$$\frac{\partial}{\partial x}\left(e^{x y}\right)$$$
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 $$$\frac{\partial}{\partial x} \left(f\left(g\right) \right)=\frac{\partial}{\partial u} \left(f\left(u\right) \right) \cdot \frac{\partial}{\partial x} \left(g \right)$$$:
$${\color{red}{\frac{\partial}{\partial x}\left(e^{x y}\right)}}={\color{red}{\frac{\partial}{\partial u}\left(e^{u}\right) \frac{\partial}{\partial x}\left(x y\right)}}$$The derivative of an exponential is $$$\frac{\partial}{\partial u} \left(e^{u} \right)=e^{u}$$$:
$${\color{red}{\frac{\partial}{\partial u}\left(e^{u}\right)}} \frac{\partial}{\partial x}\left(x y\right)={\color{red}{e^{u}}} \frac{\partial}{\partial x}\left(x y\right)$$Return to the old variable:
$$e^{{\color{red}{u}}} \frac{\partial}{\partial x}\left(x y\right)=e^{{\color{red}{x y}}} \frac{\partial}{\partial x}\left(x y\right)$$Apply the constant multiple rule $$$\frac{\partial}{\partial x} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial x} \left(f \right)$$$ with $$$c=y$$$ and $$$f=x$$$:
$$e^{x y} {\color{red}{\frac{\partial}{\partial x}\left(x y\right)}}=e^{x y} {\color{red}{y \frac{\partial}{\partial x}\left(x\right)}}$$Apply the power rule $$$\frac{\partial}{\partial x} \left(x^{n} \right)=n\cdot x^{-1+n}$$$ with $$$n=1$$$, in other words $$$\frac{\partial}{\partial x} \left(x \right)=1$$$:
$$y e^{x y} {\color{red}{\frac{\partial}{\partial x}\left(x\right)}}=y e^{x y} {\color{red}{1}}$$Thus, $$$\frac{\partial}{\partial x}\left(e^{x y}\right)=y e^{x y}$$$
Answer: $$$\frac{\partial}{\partial x}\left(e^{x y}\right)=y e^{x y}$$$