Q. 72 - Calculus

Question

Prove each statement in Exercises 71–78, using the new definition of $\ln x$as an integral and $e^x$ as the inverse of $\ln x$.

72. Prove that  $\frac{d}{dx}\left(\ln x\right)=\frac{1}{x}$.

Step-by-Step Solution

Answer

$\frac{d}{dx}\left(\ln x\right)=\frac{1}{x}$

Step 1. Given information

We have to prove that $\frac{d}{dx}\left(\ln x\right)=\frac{1}{x}$.

Step 2. Proof of the question

Let $y=\ln x$

So,

$e^y=x$

Taking derivative on both the sides,

$$\begin{gathered} \frac{dy}{dx}{e}^y=1 \\ \frac{dy}{dx}x=1 \\ \frac{d}{dx}y=\frac{1}{x} \\ \frac{d}{dx}\left(\ln x\right)=\frac{1}{x} \end{gathered}$$