No AI is used
  1. Home
  2. Calculators
  3. Calculators: Calculus III
  4. Calculus Calculator

Partial Derivative Calculator

Calculate partial derivatives step by step

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

Enter a function:

Enter the order of integration:

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 didn't work as expected, or you'd like to report an error or share feedback, please contact us.

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