Question

How do you differentiate $\dfrac{1}{2}\ln (x)$

Answer 1

The derivative is the rate of change of the quantity at some point. Now here in this question we consider the given function as y and we differentiate the given function with respect to x. Hence, we can find the derivative of the function.

Complete step-by-step solution:
Here in this question, we can find the derivative by two methods.
Method 1: In this method consider the given function as y
$y = \dfrac{1}{2}\ln (x)$
Apply the differentiation to the function
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{2}\dfrac{d}{{dx}}(\ln (x))$
We know that $\dfrac{d}{{dx}}(\ln (x)) = \dfrac{1}{x}$, applying this differentiation formula we have
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{2}.\dfrac{1}{x}$
On simplification we get
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{2x}}$

Method 2: In this method consider the given equation as y
$y = \dfrac{1}{2}\ln (x)$
Multiply the number 2 to the above equation we get
$2y = \ln (x)$
Take exponential to both sides we have
$\Rightarrow {e^{2y}} = {e^{\ln x}}$
Exponential and logarithmic functions are inverse to each other. So in the RHS we cancel the exponential number and the logarithmic number and it is written as
$\Rightarrow {e^{2y}} = x$
Applying the differentiation to the above function we have
$\Rightarrow \dfrac{d}{{dx}}({e^{2y}}) = \dfrac{d}{{dx}}(x)$
We know that $\dfrac{d}{{dx}}({e^{ax}}) = {e^{ax}}\dfrac{d}{{dx}}(ax)$, applying this differentiation formula we have
$\Rightarrow {e^{2y}}\dfrac{d}{{dx}}(2y) = \dfrac{d}{{dx}}(x)$
On differentiating we get
$\Rightarrow {e^{2y}}2.\dfrac{{dy}}{{dx}} = 1$
Substitute $y = \dfrac{1}{2}\ln (x)$ we get
$\Rightarrow {e^{2\dfrac{1}{2}\ln (x)}}2.\dfrac{{dy}}{{dx}} = 1$
On simplifying we get
$\Rightarrow {e^{\ln (x)}}2.\dfrac{{dy}}{{dx}} = 1$
Exponential and logarithmic functions are inverse to each other. So in the RHS we cancel the exponential number and the logarithmic number and it is written as
$\Rightarrow x2.\dfrac{{dy}}{{dx}} = 1$
Writing for $\dfrac{{dy}}{{dx}}$ we have
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{2x}}$
Therefore, the derivative of $\dfrac{1}{2}\ln (x)$ is $\dfrac{1}{{2x}}$
Hence by the two methods we got the answer the same.

Note: To differentiate or to find the derivative of a function we use some standard differentiation formulas. The derivative is the rate of change of quantity, in this question we differentiate the given function with respect to x and find the derivative. Exponential and logarithmic functions are inverse to each other. So we can cancel the exponential number and the logarithmic number