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편도함수 계산기
편도함수를 단계별로 계산
이 온라인 계산기는 풀이 과정을 보여 주면서 함수의 편미분을 계산합니다. 적분의 순서는 임의로 지정할 수 있습니다.
Solution
Your input: find $$$\frac{\partial}{\partial x}\left(81 x^{2} + y^{2}\right)$$$
The derivative of a sum/difference is the sum/difference of derivatives:
$${\color{red}{\frac{\partial}{\partial x}\left(81 x^{2} + y^{2}\right)}}={\color{red}{\left(\frac{\partial}{\partial x}\left(81 x^{2}\right) + \frac{\partial}{\partial x}\left(y^{2}\right)\right)}}$$Apply the constant multiple rule $$$\frac{\partial}{\partial x} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial x} \left(f \right)$$$ with $$$c=81$$$ and $$$f=x^{2}$$$:
$${\color{red}{\frac{\partial}{\partial x}\left(81 x^{2}\right)}} + \frac{\partial}{\partial x}\left(y^{2}\right)={\color{red}{\left(81 \frac{\partial}{\partial x}\left(x^{2}\right)\right)}} + \frac{\partial}{\partial x}\left(y^{2}\right)$$Apply the power rule $$$\frac{\partial}{\partial x} \left(x^{n} \right)=n\cdot x^{-1+n}$$$ with $$$n=2$$$:
$$81 {\color{red}{\frac{\partial}{\partial x}\left(x^{2}\right)}} + \frac{\partial}{\partial x}\left(y^{2}\right)=81 {\color{red}{\left(2 x^{-1 + 2}\right)}} + \frac{\partial}{\partial x}\left(y^{2}\right)=162 x + \frac{\partial}{\partial x}\left(y^{2}\right)$$The derivative of a constant is 0:
$$162 x + {\color{red}{\frac{\partial}{\partial x}\left(y^{2}\right)}}=162 x + {\color{red}{\left(0\right)}}$$Thus, $$$\frac{\partial}{\partial x}\left(81 x^{2} + y^{2}\right)=162 x$$$
Answer: $$$\frac{\partial}{\partial x}\left(81 x^{2} + y^{2}\right)=162 x$$$
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