## Calcular derivadas parciais passo a passo

Esta calculadora online calculará a derivada parcial da função, com as etapas mostradas. Você pode especificar qualquer ordem de integração.

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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).

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