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Calculadora de derivadas parciales

Calcula derivadas parciales paso a paso

Esta calculadora en línea calculará la derivada parcial de la función, mostrando los pasos. Puede especificar cualquier orden de derivación.

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 $$$\frac{\partial^{2}}{\partial y^{2}}\left(x^{2} y^{2}\right)$$$

First, find $$$\frac{\partial}{\partial y}\left(x^{2} y^{2}\right)$$$

Apply the constant multiple rule $$$\frac{\partial}{\partial y} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial y} \left(f \right)$$$ with $$$c=x^{2}$$$ and $$$f=y^{2}$$$:

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

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

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

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

Next, $$$\frac{\partial^{2}}{\partial y^{2}}\left(x^{2} y^{2}\right)=\frac{\partial}{\partial y} \left(\frac{\partial}{\partial y}\left(x^{2} y^{2}\right) \right)=\frac{\partial}{\partial y}\left(2 x^{2} y\right)$$$

Apply the constant multiple rule $$$\frac{\partial}{\partial y} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial y} \left(f \right)$$$ with $$$c=2 x^{2}$$$ and $$$f=y$$$:

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

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

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

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

Therefore, $$$\frac{\partial^{2}}{\partial y^{2}}\left(x^{2} y^{2}\right)=2 x^{2}$$$

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


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