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  1. Beranda
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  3. Kalkulator: Kalkulus III
  4. Kalkulator Kalkulus

Kalkulator Turunan Parsial

Hitung turunan parsial langkah demi langkah

Kalkulator online ini akan menghitung turunan parsial dari fungsi, beserta langkah-langkahnya. Anda dapat menentukan urutan integrasi apa pun.

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

First, find $$$\frac{\partial}{\partial x}\left(x^{2} y^{2}\right)$$$

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

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

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

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

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

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

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

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

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

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

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

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

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


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