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Calculadora de derivadas parciales

Calcular derivadas parciales paso a paso

Esta calculadora en línea calculará la derivada parcial de la función, con los pasos que se muestran. Puede especificar cualquier orden de integración.

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

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