Partial Derivative Calculator
This online calculator will calculate the partial derivative of the function, with steps shown. You can specify any order of integration.
Solution
Your input: find $$$\frac{\partial^{3}}{\partial x^{2} \partial y}\left(e^{x} + e^{y}\right)$$$
First, find $$$\frac{\partial}{\partial x}\left(e^{x} + e^{y}\right)$$$
The derivative of a sum/difference is the sum/difference of derivatives:
$$\color{red}{\frac{\partial}{\partial x}\left(e^{x} + e^{y}\right)}=\color{red}{\left(\frac{\partial}{\partial x}\left(e^{x}\right) + \frac{\partial}{\partial x}\left(e^{y}\right)\right)}$$The derivative of a constant is 0:
$$\color{red}{\frac{\partial}{\partial x}\left(e^{y}\right)} + \frac{\partial}{\partial x}\left(e^{x}\right)=\color{red}{\left(0\right)} + \frac{\partial}{\partial x}\left(e^{x}\right)$$The derivative of an exponential is $$$\frac{\partial}{\partial x} \left(e^{x} \right)=e^{x}$$$:
$$\color{red}{\frac{\partial}{\partial x}\left(e^{x}\right)}=\color{red}{e^{x}}$$Thus, $$$\frac{\partial}{\partial x}\left(e^{x} + e^{y}\right)=e^{x}$$$
Next, $$$\frac{\partial^{2}}{\partial x^{2}}\left(e^{x} + e^{y}\right)=\frac{\partial}{\partial x} \left(\frac{\partial}{\partial x}\left(e^{x} + e^{y}\right) \right)=\frac{\partial}{\partial x}\left(e^{x}\right)$$$
The derivative of an exponential is $$$\frac{\partial}{\partial x} \left(e^{x} \right)=e^{x}$$$:
$$\color{red}{\frac{\partial}{\partial x}\left(e^{x}\right)}=\color{red}{e^{x}}$$Thus, $$$\frac{\partial}{\partial x}\left(e^{x}\right)=e^{x}$$$
Next, $$$\frac{\partial^{3}}{\partial x^{2} \partial y}\left(e^{x} + e^{y}\right)=\frac{\partial}{\partial y} \left(\frac{\partial^{2}}{\partial x^{2}}\left(e^{x} + e^{y}\right) \right)=\frac{\partial}{\partial y}\left(e^{x}\right)$$$
The derivative of a constant is 0:
$$\color{red}{\frac{\partial}{\partial y}\left(e^{x}\right)}=\color{red}{\left(0\right)}$$Thus, $$$\frac{\partial}{\partial y}\left(e^{x}\right)=0$$$
Therefore, $$$\frac{\partial^{3}}{\partial x^{2} \partial y}\left(e^{x} + e^{y}\right)=0$$$
Answer: $$$\frac{\partial^{3}}{\partial x^{2} \partial y}\left(e^{x} + e^{y}\right)=0$$$