# Partial Derivative Calculator

This online calculator will calculate the partial derivative of the function, with steps shown. You can specify any order of integration.

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Hint: type x^2,y to calculate (partial^3 f)/(partial x^2 partial y), or enter x,y^2,x to find (partial^4 f)/(partial x partial y^2 partial x).

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

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: $$\frac{\partial^{3}}{\partial x^{2} \partial y}\left(e^{x} + e^{y}\right)=0$$\$