Question

Find the derivative of \[{{x}^{2}}\] with respect to log x.

Answer 1

Hint:First of all consider \[f\left( x \right)={{x}^{2}}\] and g (x) = log x. Now, find the derivation of \[{{x}^{2}}\] with respect to log x by using \[\dfrac{\dfrac{d}{dx}f\left( x \right)}{\dfrac{d}{dx}g\left( x \right)}\], substitute the values of f (x) and g (x) and use \[\dfrac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}}\text{ and }\dfrac{d}{dx}\left( \log x \right)=\dfrac{1}{x}\log e\] to get the required answer.

Complete step-by-step answer:
In this question, we have to find the derivation of \[{{x}^{2}}\] with respect to log x. If we are given two functions f (x) and g (x), then we find the derivative of f (x) with respect to g (x) by finding \[\dfrac{df\left( x \right)}{dg\left( x \right)}\]. We can write \[\dfrac{df\left( x \right)}{dg\left( x \right)}\text{ as }\dfrac{\dfrac{d}{dx}f\left( x \right)}{\dfrac{d}{dx}g\left( x \right)}\]. So, basically, we have to find the derivative of f (x) with respect to g (x). We find \[\dfrac{\text{derivative of f (x) with respect to x}}{\text{derivative of g (x) with respect to x}}\].
Now, let us consider our question. By considering \[\text{f}\left( x \right)={{x}^{2}}\text{ and }g\left( x \right)=\log x\], we get the derivation of \[f\left( x \right)={{x}^{2}}\] with respect to g (x) = log x as
\[D= \dfrac{\text{derivative of f (x) with respect to x}}{\text{derivative of g (x) with respect to x}}\]
\[\text{= }\dfrac{\dfrac{d}{dx}f\left( x \right)}{\dfrac{d}{dx}g\left( x \right)}\]
\[\text{= }\dfrac{\dfrac{d}{dx}\left( {{x}^{2}} \right)}{\dfrac{d}{dx}\left( \log x \right)}\]

We know that \[\dfrac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}}\text{ and }\dfrac{d}{dx}\left( \log x \right)=\dfrac{1}{x}\log e\]. By using this, we get,
\[D=\dfrac{\dfrac{d}{dx}\left( {{x}^{2}} \right)}{\dfrac{d}{dx}\left( \log x \right)}\]
\[D=\dfrac{2{{x}^{2-1}}}{\dfrac{1}{x}}\]
\[D=\dfrac{2x}{\dfrac{1}{x}\log e}\]
\[D=\dfrac{2{{x}^{2}}}{\log e}\]
We know that \[\dfrac{1}{{{\log }_{b}}a}={{\log }_{a}}b\]. So, we get,
\[D=2{{x}^{2}}{{\log }_{e}}10\]
So, we get the derivation of x with respect to log x as \[2{{x}^{2}}{{\log }_{e}}10\].

Note: In this question, some students find the value of \[\dfrac{dg\left( x \right)}{df\left( x \right)}\] instead of \[\dfrac{df\left( x \right)}{dg\left( x \right)}\]. So, this must be taken care of. Students must note that when we are asked for the derivation of f (x) with respect to g (x), we use \[\dfrac{df\left( x \right)}{dg\left( x \right)}\] and when we are asked for the derivation of g (x) with respect of f (x), we use \[\dfrac{dg\left( x \right)}{df\left( x \right)}\]. Here, students can cross-check their answer by integrating \[2{{x}^{2}}{{\log }_{e}}10\] and check if it is equal to initial expression or not.