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).

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