Question

How do you find the derivative of $\ln (4x)$?

Answer 1

Start differentiating the expression in the question by using the formula $\dfrac{d}{{dx}}(\ln (x)) = \dfrac{1}{x}$. As the term involved is 4x, differentiate that with respect to x too and simplify this expression.

Complete Step by Step Solution:
We have to find the derivative of $\ln (4x)$.
Let $y = \ln (4x)$ .
Differentiating both sides of the equation with respect to x
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{4x}}\dfrac{d}{{dx}}(4x)$
$(\because$ the derivative of $\ln (x) = \dfrac{1}{x}$.
However, an important thing to be taken into consideration is - the term involved is 4x. So we must also differentiate 4x with respect to x.)
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{4x}} \times 4$
Simplifying
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{x}$

$\therefore \dfrac{{dy}}{{dx}} = \dfrac{1}{x}$ is the derivative of $\ln (4x)$.

Note:
Alternate method:
We have to find the derivative of $\ln (4x)$.
Let $y = \ln (4x)$.
Let $4x = t$
Differentiating both sides of the equation with respect to x,
$\Rightarrow 4\dfrac{d}{{dx}}(x) = \dfrac{{dt}}{{dx}}$
$\Rightarrow \dfrac{{dt}}{{dx}} = 4$
Let $\dfrac{{dt}}{{dx}} = 4$ be labelled as equation 1.
Substituting the value of 4x in the expression, $y = \ln (4x)$ we get $y = \ln (t)$
Differentiating both sides of the above equation with respect to x,
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{t}\dfrac{{dt}}{{dx}}$
$(\because$ the derivative of $\ln (t) = \dfrac{1}{t}$.
As the equation involving the variable t is differentiated with respect to x, we need to differentiate t with respect to x too. Thus we obtain the derivative.)
Substituting the value of $\dfrac{{dt}}{{dx}}$ obtained in equation 1 in the above expression, we get $\dfrac{{dy}}{{dx}} = \dfrac{1}{t}(4)$.
Now we replace the value of t with 4x and change the variable terms back to x.
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{4}{{4x}}$
$\therefore \dfrac{{dy}}{{dx}} = \dfrac{1}{x}$