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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 $$$\frac{\partial}{\partial y}\left(3 x + 4 y\right)$$$

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

$${\color{red}{\frac{\partial}{\partial y}\left(3 x + 4 y\right)}}={\color{red}{\left(\frac{\partial}{\partial y}\left(3 x\right) + \frac{\partial}{\partial y}\left(4 y\right)\right)}}$$

Apply the constant multiple rule $$$\frac{\partial}{\partial y} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial y} \left(f \right)$$$ with $$$c=4$$$ and $$$f=y$$$:

$${\color{red}{\frac{\partial}{\partial y}\left(4 y\right)}} + \frac{\partial}{\partial y}\left(3 x\right)={\color{red}{\left(4 \frac{\partial}{\partial y}\left(y\right)\right)}} + \frac{\partial}{\partial y}\left(3 x\right)$$

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

$$4 {\color{red}{\frac{\partial}{\partial y}\left(y\right)}} + \frac{\partial}{\partial y}\left(3 x\right)=4 {\color{red}{1}} + \frac{\partial}{\partial y}\left(3 x\right)$$

The derivative of a constant is 0:

$$4 + {\color{red}{\frac{\partial}{\partial y}\left(3 x\right)}}=4 + {\color{red}{\left(0\right)}}$$

Thus, $$$\frac{\partial}{\partial y}\left(3 x + 4 y\right)=4$$$

Answer: $$$\frac{\partial}{\partial y}\left(3 x + 4 y\right)=4$$$