Question

what is the differentiatio of logx with respect to X^2?
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Answer 1

Let,

$\longrightarrow u=\log x$

Differentiating wrt $x,$

$\longrightarrow \dfrac{du}{dx}=\dfrac{d}{dx}(\log x)$

$\longrightarrow \dfrac{du}{dx}=\dfrac{1}{x}\quad\quad\dots(1)$

Let,

$\longrightarrow v=x^2$

Differentiating wrt $x,$

$\longrightarrow \dfrac{dv}{dx}=\dfrac{d}{dx}(x^2)$

$\longrightarrow \dfrac{dv}{dx}=2x\quad\quad\dots(2)$

Dividing (1) by (2),

$\longrightarrow \dfrac{du}{dx}\div\dfrac{dv}{dx}=\dfrac{1}{x}\div 2x$

$\longrightarrow \dfrac{du}{dx}\times\dfrac{dx}{dv}=\dfrac{1}{x\cdot 2x}$

$\longrightarrow\underline{\underline{\dfrac{du}{dv}=\dfrac{1}{2x^2}}}$

This is the derivative of $\log x$ wrt $x^2.$