Question

What is derivative of $\ln \left( 8x \right)$ ?

Answer 1

Here in this question we have been asked to find the value of the derivative of the given logarithmic function $\ln \left( 8x \right)$. Firstly we will assume $8x=u$ and then we will find the derivative of the given expression using the basic rule $\dfrac{dy}{du}\dfrac{du}{dx}=\dfrac{dy}{dx}$ where $y$ is a function of $u$ and $u$ is a function of $x$ . We also know that $\dfrac{d}{dx}\ln x=\dfrac{1}{x}$.

Complete step-by-step solution:
Now considering from the question we have been asked to find the value of the derivative of the given logarithmic function $\ln \left( 8x \right)$.
From the basic concepts of the derivatives we know the basic rule given as $\dfrac{dy}{du}\dfrac{du}{dx}=\dfrac{dy}{dx}$ where $y$ is a function of $u$ and $u$ is a function of $x$ .
We also know that $\dfrac{d}{dx}\ln x=\dfrac{1}{x}$.
Let us assume that $8x=u$ and simplify the given expression. By using our assumption we will have $\Rightarrow \dfrac{d}{dx}\ln \left( 8x \right)=\dfrac{d}{dx}\ln \left( u \right)$ .
Now by using the basic rule $\dfrac{dy}{du}\dfrac{du}{dx}=\dfrac{dy}{dx}$ we will have $\Rightarrow \dfrac{d}{dx}\ln \left( u \right)=\dfrac{d}{du}\ln \left( u \right)\dfrac{du}{dx}$ .
Now by further simplifying this expression using $\dfrac{d}{dx}\ln x=\dfrac{1}{x}$ we will have $\Rightarrow \dfrac{d}{du}\ln \left( u \right)\dfrac{du}{dx}=\dfrac{1}{u}\dfrac{du}{dx}$ .
Now by replacing $u=8x$ we will get $\Rightarrow \dfrac{1}{u}\dfrac{du}{dx}=\dfrac{1}{8x}\dfrac{d\left( 8x \right)}{dx}$.
Finally we will end up having
$\begin{aligned} & \Rightarrow \dfrac{1}{8x}\dfrac{d\left( 8x \right)}{dx}=\dfrac{1}{8x}8 \\\ & \Rightarrow \dfrac{1}{x} \\\ \end{aligned}$
Therefore we conclude that the value of the derivative of $\ln \left( 8x \right)$ is given as $\dfrac{1}{x}$.

Note: During the process of answering questions of this type we should be sure with the concepts that we are going to apply in between the steps. We can derive a simple conclusion from this question as follows $\dfrac{d}{dx}\ln \left( ax \right)=\dfrac{1}{x}$ where $a$ can be any integer and simple answer any other questions of this type using this conclusion.