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  1. 홈
  2. 계산기
  3. 계산기: 미적분학 III
  4. 미적분 계산기

편도함수 계산기

편도함수를 단계별로 계산

이 온라인 계산기는 풀이 과정을 보여 주면서 함수의 편미분을 계산합니다. 적분의 순서는 임의로 지정할 수 있습니다.

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 $$$\frac{\partial}{\partial z}\left(x y^{2} z^{3}\right)$$$

Apply the constant multiple rule $$$\frac{\partial}{\partial z} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial z} \left(f \right)$$$ with $$$c=x y^{2}$$$ and $$$f=z^{3}$$$:

$${\color{red}{\frac{\partial}{\partial z}\left(x y^{2} z^{3}\right)}}={\color{red}{x y^{2} \frac{\partial}{\partial z}\left(z^{3}\right)}}$$

Apply the power rule $$$\frac{\partial}{\partial z} \left(z^{n} \right)=n\cdot z^{-1+n}$$$ with $$$n=3$$$:

$$x y^{2} {\color{red}{\frac{\partial}{\partial z}\left(z^{3}\right)}}=x y^{2} {\color{red}{\left(3 z^{-1 + 3}\right)}}=3 x y^{2} z^{2}$$

Thus, $$$\frac{\partial}{\partial z}\left(x y^{2} z^{3}\right)=3 x y^{2} z^{2}$$$

Answer: $$$\frac{\partial}{\partial z}\left(x y^{2} z^{3}\right)=3 x y^{2} z^{2}$$$


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