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偏導數計算器

逐步計算偏導數

此線上計算器可計算函數的偏導數，並顯示步驟。您可以指定任意的積分順序。

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)`.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please contact us.

Solution

Your input: find $$$\frac{\partial}{\partial y}\left(4 x + y\right)$$$

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

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

The derivative of a constant is 0:

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

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

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

Thus, $$$\frac{\partial}{\partial y}\left(4 x + y\right)=1$$$

Answer: $$$\frac{\partial}{\partial y}\left(4 x + y\right)=1$$$


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