Question

How do you find the derivative of $\ln \left( {1 + {x^2}} \right)$.

Answer 1

Hint: This is a derivative of the composite function $f\left( {g\left( x \right)} \right)$. We can calculate the derivative of this composite function with the help of following

Formula Used:
$\dfrac{d}{{dx}}f\left( {g\left( x \right)} \right) = f'\left( {g\left( x \right)} \right) \cdot g'\left( x \right)$

Complete step by step solution:
In the derivative of composite function $f\left( {g\left( x \right)} \right)$ the two functions are as follows:
$
  f\left( x \right) = \ln \left( x \right) \\
  g\left( x \right) = {x^2} + 1 \\
 $
The chain formula of deriving the derivative of the given function is
$\dfrac{d}{{dx}}f\left( {g\left( x \right)} \right) = f'\left( {g\left( x \right)} \right) \cdot g'\left( x \right) \cdots \cdots \left( 1 \right)$
Here,
$f\left( x \right) = \ln \left( x \right)$
Now the derivative of this function is
$f'\left( x \right) = \dfrac{1}{x}$
And another function
$g\left( x \right) = 1 + {x^2}$
And the derivative of this function is
$g'\left( x \right) = 2x$
Therefore, we can calculate the derivative of whole function by the following expression
$f'\left( {g\left( x \right)} \right) = \dfrac{1}{{g\left( x \right)}}$
Now substitute the value of $g\left( x \right)$in the above expression
$f'\left( {g\left( x \right)} \right) = \dfrac{1}{{{x^2} + 1}}$
Now substitute all the values in equation (1) we get

$\dfrac{d}{{dx}}\left[ {\ln \left( {{x^2} + 1} \right)} \right] = \dfrac{{2x}}{{1 + {x^2}}}$
This is the final solution of the given problem.

Note:
We can also find out the derivative of logarithmic functions with the help of the following method.
Let’s suppose
$y = \ln u$
Where $u$ is the function of $x$
Now differentiate the above equation with respect to $x$ we get
$\dfrac{{dy}}{{dx}} = \dfrac{1}{u}\dfrac{{du}}{{dx}} \cdots \cdots \left( 2 \right)$
And if $u$ is function of $x$ then
$u = f\left( x \right)$
Then in that case the equation (2) can be written in the following form
$\dfrac{{dy}}{{dx}} = \dfrac{{f'\left( x \right)}}{{f\left( x \right)}}$
While calculating the derivatives of the logarithmic function it is important to know all the formulas of derivatives and we have to first differentiate the given log function then the inside function.