Question

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

Answer 1

Hint: 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