Question

Which one of the following is the differentiation of $x$ with respect to $\log x,x > 0$:
A. $\dfrac{1}{{{x^2}}}$
B. $x$
C. $\log x$
D. ${e^x}$

Answer 1

In the given problem, we are required to differentiate x with respect to $\log x$ for $x > 0$. Since, we have to differentiate a function with respect to another function, we first differentiate both the functions with respect to a simple variable and then divide the results in order to get the final answer. So, we apply the chain rule of differentiation in the process of differentiating x with respect to $\log x$. Derivative of $\log x$ with respect to x must be remembered in order to solve the given problem.

Complete step by step answer:
We have to find the derivative of x with respect to $\log x$. So, we have, $\dfrac{{dx}}{{d\left( {\log x} \right)}}$. Now, we find the derivative of both the functions with respect to x in numerator and denominator. So, we get,
$\Rightarrow \dfrac{{\dfrac{{dx}}{{dx}}}}{{\dfrac{{d\left( {\log x} \right)}}{{dx}}}}$
Now, we know that the derivative of x with respect to x is $1$ and the derivative of logarithmic function with respect to x is $\left( {\dfrac{1}{x}} \right)$. Hence, we get,
$\Rightarrow \dfrac{1}{{\left( {\dfrac{1}{x}} \right)}}$
So, simplifying the expression, we get,
$\Rightarrow x$

So, the derivative of x with respect to $\log x$ is $x$, for $x > 0$.

Note: The derivatives of basic and simple functions such as $\log x$ must be learned by heart in order to find derivatives of complex composite functions using chain rule of differentiation. The chain rule of differentiation involves differentiating a composite by introducing new unknowns to ease the process and examine the behaviour of function layer by layer. Whenever we have to find the derivative of a function with respect to another function, we first find the derivative of both the functions with the variable so as to ease the process of differentiation and then divide both the results.