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