## 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 $\frac{\partial}{\partial y}\left(x^{2} + y^{2} + z^{2} - 14\right)$

The derivative of a sum/difference is the sum/difference of derivatives:

$${\color{red}{\frac{\partial}{\partial y}\left(x^{2} + y^{2} + z^{2} - 14\right)}}={\color{red}{\left(- \frac{\partial}{\partial y}\left(14\right) + \frac{\partial}{\partial y}\left(x^{2}\right) + \frac{\partial}{\partial y}\left(y^{2}\right) + \frac{\partial}{\partial y}\left(z^{2}\right)\right)}}$$

The derivative of a constant is 0:

$${\color{red}{\frac{\partial}{\partial y}\left(z^{2}\right)}} - \frac{\partial}{\partial y}\left(14\right) + \frac{\partial}{\partial y}\left(x^{2}\right) + \frac{\partial}{\partial y}\left(y^{2}\right)={\color{red}{\left(0\right)}} - \frac{\partial}{\partial y}\left(14\right) + \frac{\partial}{\partial y}\left(x^{2}\right) + \frac{\partial}{\partial y}\left(y^{2}\right)$$

The derivative of a constant is 0:

$$- {\color{red}{\frac{\partial}{\partial y}\left(14\right)}} + \frac{\partial}{\partial y}\left(x^{2}\right) + \frac{\partial}{\partial y}\left(y^{2}\right)=- {\color{red}{\left(0\right)}} + \frac{\partial}{\partial y}\left(x^{2}\right) + \frac{\partial}{\partial y}\left(y^{2}\right)$$

The derivative of a constant is 0:

$${\color{red}{\frac{\partial}{\partial y}\left(x^{2}\right)}} + \frac{\partial}{\partial y}\left(y^{2}\right)={\color{red}{\left(0\right)}} + \frac{\partial}{\partial y}\left(y^{2}\right)$$

Apply the power rule $\frac{\partial}{\partial y} \left(y^{n} \right)=n\cdot y^{-1+n}$ with $n=2$:

$${\color{red}{\frac{\partial}{\partial y}\left(y^{2}\right)}}={\color{red}{\left(2 y^{-1 + 2}\right)}}=2 y$$

Thus, $\frac{\partial}{\partial y}\left(x^{2} + y^{2} + z^{2} - 14\right)=2 y$

Answer: $\frac{\partial}{\partial y}\left(x^{2} + y^{2} + z^{2} - 14\right)=2 y$