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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.

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)`.

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Solution

Your input: find $$$\frac{\partial}{\partial z}\left(x^{2} + y^{2} + z^{2} - 14\right)$$$

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

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

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

$${\color{red}{\frac{\partial}{\partial z}\left(z^{2}\right)}} - \frac{\partial}{\partial z}\left(14\right) + \frac{\partial}{\partial z}\left(x^{2}\right) + \frac{\partial}{\partial z}\left(y^{2}\right)={\color{red}{\left(2 z^{-1 + 2}\right)}} - \frac{\partial}{\partial z}\left(14\right) + \frac{\partial}{\partial z}\left(x^{2}\right) + \frac{\partial}{\partial z}\left(y^{2}\right)=2 z - \frac{\partial}{\partial z}\left(14\right) + \frac{\partial}{\partial z}\left(x^{2}\right) + \frac{\partial}{\partial z}\left(y^{2}\right)$$

The derivative of a constant is 0:

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

The derivative of a constant is 0:

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

The derivative of a constant is 0:

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

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

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