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Kalkylator för partialderivator
Beräkna partiella derivator steg för steg
Den här onlinekalkylatorn beräknar funktionens partialderivata och visar stegen. Du kan ange vilken integrationsordning som helst.
Solution
Your input: find $$$\frac{\partial^{2}}{\partial x^{2}}\left(x^{3} + 4 x y^{2} + 5 y^{3} - 10\right)$$$
First, find $$$\frac{\partial}{\partial x}\left(x^{3} + 4 x y^{2} + 5 y^{3} - 10\right)$$$
The derivative of a sum/difference is the sum/difference of derivatives:
$${\color{red}{\frac{\partial}{\partial x}\left(x^{3} + 4 x y^{2} + 5 y^{3} - 10\right)}}={\color{red}{\left(- \frac{\partial}{\partial x}\left(10\right) + \frac{\partial}{\partial x}\left(x^{3}\right) + \frac{\partial}{\partial x}\left(5 y^{3}\right) + \frac{\partial}{\partial x}\left(4 x y^{2}\right)\right)}}$$Apply the constant multiple rule $$$\frac{\partial}{\partial x} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial x} \left(f \right)$$$ with $$$c=4 y^{2}$$$ and $$$f=x$$$:
$${\color{red}{\frac{\partial}{\partial x}\left(4 x y^{2}\right)}} - \frac{\partial}{\partial x}\left(10\right) + \frac{\partial}{\partial x}\left(x^{3}\right) + \frac{\partial}{\partial x}\left(5 y^{3}\right)={\color{red}{4 y^{2} \frac{\partial}{\partial x}\left(x\right)}} - \frac{\partial}{\partial x}\left(10\right) + \frac{\partial}{\partial x}\left(x^{3}\right) + \frac{\partial}{\partial x}\left(5 y^{3}\right)$$Apply the power rule $$$\frac{\partial}{\partial x} \left(x^{n} \right)=n\cdot x^{-1+n}$$$ with $$$n=1$$$, in other words $$$\frac{\partial}{\partial x} \left(x \right)=1$$$:
$$4 y^{2} {\color{red}{\frac{\partial}{\partial x}\left(x\right)}} - \frac{\partial}{\partial x}\left(10\right) + \frac{\partial}{\partial x}\left(x^{3}\right) + \frac{\partial}{\partial x}\left(5 y^{3}\right)=4 y^{2} {\color{red}{1}} - \frac{\partial}{\partial x}\left(10\right) + \frac{\partial}{\partial x}\left(x^{3}\right) + \frac{\partial}{\partial x}\left(5 y^{3}\right)$$The derivative of a constant is 0:
$$4 y^{2} - {\color{red}{\frac{\partial}{\partial x}\left(10\right)}} + \frac{\partial}{\partial x}\left(x^{3}\right) + \frac{\partial}{\partial x}\left(5 y^{3}\right)=4 y^{2} - {\color{red}{\left(0\right)}} + \frac{\partial}{\partial x}\left(x^{3}\right) + \frac{\partial}{\partial x}\left(5 y^{3}\right)$$Apply the power rule $$$\frac{\partial}{\partial x} \left(x^{n} \right)=n\cdot x^{-1+n}$$$ with $$$n=3$$$:
$$4 y^{2} + {\color{red}{\frac{\partial}{\partial x}\left(x^{3}\right)}} + \frac{\partial}{\partial x}\left(5 y^{3}\right)=4 y^{2} + {\color{red}{\left(3 x^{-1 + 3}\right)}} + \frac{\partial}{\partial x}\left(5 y^{3}\right)=3 x^{2} + 4 y^{2} + \frac{\partial}{\partial x}\left(5 y^{3}\right)$$The derivative of a constant is 0:
$$3 x^{2} + 4 y^{2} + {\color{red}{\frac{\partial}{\partial x}\left(5 y^{3}\right)}}=3 x^{2} + 4 y^{2} + {\color{red}{\left(0\right)}}$$Thus, $$$\frac{\partial}{\partial x}\left(x^{3} + 4 x y^{2} + 5 y^{3} - 10\right)=3 x^{2} + 4 y^{2}$$$
Next, $$$\frac{\partial^{2}}{\partial x^{2}}\left(x^{3} + 4 x y^{2} + 5 y^{3} - 10\right)=\frac{\partial}{\partial x} \left(\frac{\partial}{\partial x}\left(x^{3} + 4 x y^{2} + 5 y^{3} - 10\right) \right)=\frac{\partial}{\partial x}\left(3 x^{2} + 4 y^{2}\right)$$$
The derivative of a sum/difference is the sum/difference of derivatives:
$${\color{red}{\frac{\partial}{\partial x}\left(3 x^{2} + 4 y^{2}\right)}}={\color{red}{\left(\frac{\partial}{\partial x}\left(3 x^{2}\right) + \frac{\partial}{\partial x}\left(4 y^{2}\right)\right)}}$$The derivative of a constant is 0:
$${\color{red}{\frac{\partial}{\partial x}\left(4 y^{2}\right)}} + \frac{\partial}{\partial x}\left(3 x^{2}\right)={\color{red}{\left(0\right)}} + \frac{\partial}{\partial x}\left(3 x^{2}\right)$$Apply the constant multiple rule $$$\frac{\partial}{\partial x} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial x} \left(f \right)$$$ with $$$c=3$$$ and $$$f=x^{2}$$$:
$${\color{red}{\frac{\partial}{\partial x}\left(3 x^{2}\right)}}={\color{red}{\left(3 \frac{\partial}{\partial x}\left(x^{2}\right)\right)}}$$Apply the power rule $$$\frac{\partial}{\partial x} \left(x^{n} \right)=n\cdot x^{-1+n}$$$ with $$$n=2$$$:
$$3 {\color{red}{\frac{\partial}{\partial x}\left(x^{2}\right)}}=3 {\color{red}{\left(2 x^{-1 + 2}\right)}}=6 x$$Thus, $$$\frac{\partial}{\partial x}\left(3 x^{2} + 4 y^{2}\right)=6 x$$$
Therefore, $$$\frac{\partial^{2}}{\partial x^{2}}\left(x^{3} + 4 x y^{2} + 5 y^{3} - 10\right)=6 x$$$
Answer: $$$\frac{\partial^{2}}{\partial x^{2}}\left(x^{3} + 4 x y^{2} + 5 y^{3} - 10\right)=6 x$$$