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.
Solution
Your input: find $$$\frac{\partial^{2}}{\partial y^{2}}\left(x^{4} - 4 x y + y^{4} + 1\right)$$$
First, find $$$\frac{\partial}{\partial y}\left(x^{4} - 4 x y + y^{4} + 1\right)$$$
The derivative of a sum/difference is the sum/difference of derivatives:
$${\color{red}{\frac{\partial}{\partial y}\left(x^{4} - 4 x y + y^{4} + 1\right)}}={\color{red}{\left(\frac{\partial}{\partial y}\left(1\right) + \frac{\partial}{\partial y}\left(x^{4}\right) + \frac{\partial}{\partial y}\left(y^{4}\right) - \frac{\partial}{\partial y}\left(4 x y\right)\right)}}$$Apply the power rule $$$\frac{\partial}{\partial y} \left(y^{n} \right)=n\cdot y^{-1+n}$$$ with $$$n=4$$$:
$${\color{red}{\frac{\partial}{\partial y}\left(y^{4}\right)}} + \frac{\partial}{\partial y}\left(1\right) + \frac{\partial}{\partial y}\left(x^{4}\right) - \frac{\partial}{\partial y}\left(4 x y\right)={\color{red}{\left(4 y^{-1 + 4}\right)}} + \frac{\partial}{\partial y}\left(1\right) + \frac{\partial}{\partial y}\left(x^{4}\right) - \frac{\partial}{\partial y}\left(4 x y\right)=4 y^{3} + \frac{\partial}{\partial y}\left(1\right) + \frac{\partial}{\partial y}\left(x^{4}\right) - \frac{\partial}{\partial y}\left(4 x y\right)$$Apply the constant multiple rule $$$\frac{\partial}{\partial y} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial y} \left(f \right)$$$ with $$$c=4 x$$$ and $$$f=y$$$:
$$4 y^{3} - {\color{red}{\frac{\partial}{\partial y}\left(4 x y\right)}} + \frac{\partial}{\partial y}\left(1\right) + \frac{\partial}{\partial y}\left(x^{4}\right)=4 y^{3} - {\color{red}{4 x \frac{\partial}{\partial y}\left(y\right)}} + \frac{\partial}{\partial y}\left(1\right) + \frac{\partial}{\partial y}\left(x^{4}\right)$$Apply the power rule $$$\frac{\partial}{\partial y} \left(y^{n} \right)=n\cdot y^{-1+n}$$$ with $$$n=1$$$, in other words $$$\frac{\partial}{\partial y} \left(y \right)=1$$$:
$$- 4 x {\color{red}{\frac{\partial}{\partial y}\left(y\right)}} + 4 y^{3} + \frac{\partial}{\partial y}\left(1\right) + \frac{\partial}{\partial y}\left(x^{4}\right)=- 4 x {\color{red}{1}} + 4 y^{3} + \frac{\partial}{\partial y}\left(1\right) + \frac{\partial}{\partial y}\left(x^{4}\right)$$The derivative of a constant is 0:
$$- 4 x + 4 y^{3} + {\color{red}{\frac{\partial}{\partial y}\left(1\right)}} + \frac{\partial}{\partial y}\left(x^{4}\right)=- 4 x + 4 y^{3} + {\color{red}{\left(0\right)}} + \frac{\partial}{\partial y}\left(x^{4}\right)$$The derivative of a constant is 0:
$$- 4 x + 4 y^{3} + {\color{red}{\frac{\partial}{\partial y}\left(x^{4}\right)}}=- 4 x + 4 y^{3} + {\color{red}{\left(0\right)}}$$Thus, $$$\frac{\partial}{\partial y}\left(x^{4} - 4 x y + y^{4} + 1\right)=- 4 x + 4 y^{3}$$$
Next, $$$\frac{\partial^{2}}{\partial y^{2}}\left(x^{4} - 4 x y + y^{4} + 1\right)=\frac{\partial}{\partial y} \left(\frac{\partial}{\partial y}\left(x^{4} - 4 x y + y^{4} + 1\right) \right)=\frac{\partial}{\partial y}\left(- 4 x + 4 y^{3}\right)$$$
The derivative of a sum/difference is the sum/difference of derivatives:
$${\color{red}{\frac{\partial}{\partial y}\left(- 4 x + 4 y^{3}\right)}}={\color{red}{\left(- \frac{\partial}{\partial y}\left(4 x\right) + \frac{\partial}{\partial y}\left(4 y^{3}\right)\right)}}$$Apply the constant multiple rule $$$\frac{\partial}{\partial y} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial y} \left(f \right)$$$ with $$$c=4$$$ and $$$f=y^{3}$$$:
$${\color{red}{\frac{\partial}{\partial y}\left(4 y^{3}\right)}} - \frac{\partial}{\partial y}\left(4 x\right)={\color{red}{\left(4 \frac{\partial}{\partial y}\left(y^{3}\right)\right)}} - \frac{\partial}{\partial y}\left(4 x\right)$$Apply the power rule $$$\frac{\partial}{\partial y} \left(y^{n} \right)=n\cdot y^{-1+n}$$$ with $$$n=3$$$:
$$4 {\color{red}{\frac{\partial}{\partial y}\left(y^{3}\right)}} - \frac{\partial}{\partial y}\left(4 x\right)=4 {\color{red}{\left(3 y^{-1 + 3}\right)}} - \frac{\partial}{\partial y}\left(4 x\right)=12 y^{2} - \frac{\partial}{\partial y}\left(4 x\right)$$The derivative of a constant is 0:
$$12 y^{2} - {\color{red}{\frac{\partial}{\partial y}\left(4 x\right)}}=12 y^{2} - {\color{red}{\left(0\right)}}$$Thus, $$$\frac{\partial}{\partial y}\left(- 4 x + 4 y^{3}\right)=12 y^{2}$$$
Therefore, $$$\frac{\partial^{2}}{\partial y^{2}}\left(x^{4} - 4 x y + y^{4} + 1\right)=12 y^{2}$$$
Answer: $$$\frac{\partial^{2}}{\partial y^{2}}\left(x^{4} - 4 x y + y^{4} + 1\right)=12 y^{2}$$$