# Partial Derivative Calculator

## Calculate partial derivatives step by step

This online calculator will calculate the partial derivative of the function, with steps shown. You can specify any order of integration.

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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 $\frac{\partial}{\partial x}\left(81 x^{2} + y^{2}\right)$

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

$$\color{red}{\frac{\partial}{\partial x}\left(81 x^{2} + y^{2}\right)}=\color{red}{\left(\frac{\partial}{\partial x}\left(81 x^{2}\right) + \frac{\partial}{\partial x}\left(y^{2}\right)\right)}$$

Apply the constant multiple rule $\frac{\partial}{\partial x} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial x} \left(f \right)$ with $c=81$ and $f=x^{2}$:

$$\color{red}{\frac{\partial}{\partial x}\left(81 x^{2}\right)} + \frac{\partial}{\partial x}\left(y^{2}\right)=\color{red}{\left(81 \frac{\partial}{\partial x}\left(x^{2}\right)\right)} + \frac{\partial}{\partial x}\left(y^{2}\right)$$

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

$$81 \color{red}{\frac{\partial}{\partial x}\left(x^{2}\right)} + \frac{\partial}{\partial x}\left(y^{2}\right)=81 \color{red}{\left(2 x^{-1 + 2}\right)} + \frac{\partial}{\partial x}\left(y^{2}\right)=162 x + \frac{\partial}{\partial x}\left(y^{2}\right)$$

The derivative of a constant is 0:

$$162 x + \color{red}{\frac{\partial}{\partial x}\left(y^{2}\right)}=162 x + \color{red}{\left(0\right)}$$

Thus, $\frac{\partial}{\partial x}\left(81 x^{2} + y^{2}\right)=162 x$

Answer: $\frac{\partial}{\partial x}\left(81 x^{2} + y^{2}\right)=162 x$