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 $\frac{\partial^{5}}{\partial l \partial a \partial m \partial d \partial a}\left(81 x^{2} + y^{2}\right)$

First, find $\frac{\partial}{\partial l}\left(81 x^{2} + y^{2}\right)$

The derivative of a constant is 0:

$$\color{red}{\frac{\partial}{\partial l}\left(81 x^{2} + y^{2}\right)}=\color{red}{\left(0\right)}$$

Thus, $\frac{\partial}{\partial l}\left(81 x^{2} + y^{2}\right)=0$

Next, $\frac{\partial^{2}}{\partial l \partial a}\left(81 x^{2} + y^{2}\right)=\frac{\partial}{\partial a} \left(\frac{\partial}{\partial l}\left(81 x^{2} + y^{2}\right) \right)=\frac{\partial}{\partial a}\left(0\right)$

The derivative of a constant is 0:

$$\color{red}{\frac{\partial}{\partial a}\left(0\right)}=\color{red}{\left(0\right)}$$

Thus, $\frac{\partial}{\partial a}\left(0\right)=0$

Next, $\frac{\partial^{3}}{\partial l \partial a \partial m}\left(81 x^{2} + y^{2}\right)=\frac{\partial}{\partial m} \left(\frac{\partial^{2}}{\partial l \partial a}\left(81 x^{2} + y^{2}\right) \right)=\frac{\partial}{\partial m}\left(0\right)$

The derivative of a constant is 0:

$$\color{red}{\frac{\partial}{\partial m}\left(0\right)}=\color{red}{\left(0\right)}$$

Thus, $\frac{\partial}{\partial m}\left(0\right)=0$

Next, $\frac{\partial^{4}}{\partial l \partial a \partial m \partial d}\left(81 x^{2} + y^{2}\right)=\frac{\partial}{\partial d} \left(\frac{\partial^{3}}{\partial l \partial a \partial m}\left(81 x^{2} + y^{2}\right) \right)=\frac{\partial}{\partial d}\left(0\right)$

The derivative of a constant is 0:

$$\color{red}{\frac{\partial}{\partial d}\left(0\right)}=\color{red}{\left(0\right)}$$

Thus, $\frac{\partial}{\partial d}\left(0\right)=0$

Next, $\frac{\partial^{5}}{\partial l \partial a \partial m \partial d \partial a}\left(81 x^{2} + y^{2}\right)=\frac{\partial}{\partial a} \left(\frac{\partial^{4}}{\partial l \partial a \partial m \partial d}\left(81 x^{2} + y^{2}\right) \right)=\frac{\partial}{\partial a}\left(0\right)$

The derivative of a constant is 0:

$$\color{red}{\frac{\partial}{\partial a}\left(0\right)}=\color{red}{\left(0\right)}$$

Thus, $\frac{\partial}{\partial a}\left(0\right)=0$

Therefore, $\frac{\partial^{5}}{\partial l \partial a \partial m \partial d \partial a}\left(81 x^{2} + y^{2}\right)=0$

Answer: $\frac{\partial^{5}}{\partial l \partial a \partial m \partial d \partial a}\left(81 x^{2} + y^{2}\right)=0$