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  4. 微積分計算機

偏微分計算機

偏導関数を段階的に計算する

このオンライン計算機は、関数の偏導関数を計算し、計算手順も表示します。積分の順序は任意に指定できます。

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

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

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

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

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

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

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

The derivative of a constant is 0:

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

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

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


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