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.