Question

What is the differentiation of $\log 2x$ ?

Answer 1

Hint: In the given problem, we are required to differentiate $\log 2x$ with respect to x. Since, $\log 2x$ is a composite function, we will have to apply the chain rule of differentiation in the process of differentiating $\log 2x$ . So, differentiation of $\log 2x$ with respect to x will be done layer by layer using the chain rule of differentiation. Also derivatives of 2x with respect to x must be remembered in order to solve the given problem.

Complete step by step solution:
So, we have, ${\displaystyle \frac{d}{dx}}\left(\log 2x\right)$
Keeping the expression inside the logarithmic function inside the bracket, we get,
$=$${\displaystyle \frac{d}{dx}}\left(\log \left(2x\right)\right)$
Now, Let us assume $u=2x$. So substituting $2x$ as $u$, we get,
$=$${\displaystyle \frac{d}{dx}}\left(\log u\right)$
Now, we know that differentiation of logarithmic function $\log x$ with respect to x is $\left({\displaystyle \frac{1}{x}}\right)$. So, we get,
$=$${\displaystyle \frac{1}{u}}\times {\displaystyle \frac{du}{dx}}$
Now, putting back $u$as $2x$, we get,
$=$${\displaystyle \frac{1}{2x}}\times {\displaystyle \frac{d\left(2x\right)}{dx}}$ because $${\displaystyle \frac{du}{dx}}={\displaystyle \frac{d\left(2x\right)}{dx}}$$
Now, we take the constants out of the differentiation. So, we get,
$=$${\displaystyle \frac{2}{2x}}\times {\displaystyle \frac{d\left(x\right)}{dx}}$
Now, we know that the derivative of x with respect to x is $1$. Hence, we get,
$=$${\displaystyle \frac{2}{2x}}\times 1$
Cancelling the common factors in numerator and denominator, we get,
$=$${\displaystyle \frac{1}{x}}$
So, the derivative of $\log 2x$ with respect to x is ${\displaystyle \frac{1}{x}}$.
So, the correct answer is “${\displaystyle \frac{1}{x}}$”.

Note: 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. The derivative of the basic logarithmic function $\log x$ with respect to x is ${\displaystyle \frac{1}{x}}$. The power rule of differentiation is as follows: ${\displaystyle \frac{d\left(x^n\right)}{dx}}=n{x}^{n-1}$.