Question

How to differentiate $\ln \left( {{x^2} + 1} \right)$

Answer 1

Hint : There are two different formulas you need to use to solve this problem. One is the basic differentiation formula and the other one is to differentiate ln. The basic differentiation formula is ${x^n} = n{x^{n - 1}}$ and then if we differentiate $\ln x$ we get, $\dfrac{1}{x}$ . Keep this in mind and solve the problem.

Complete step-by-step answer :
Here we have more than one function, so we consider this below mentioned rule, which is known as chain rule. Here we take $f(g(x))$ as $\ln ({x^2} + 1)$ . First differentiate $\ln ({x^2} + 1)$ and then differentiate ${x^2} + 1$ separately. When we differentiate $\ln ({x^2} + 1)$ we get, $\dfrac{1}{{{x^2} + 1}}$ and when we differentiate $({x^2} + 1)$ we get $(2x + 0)$ , as we know if we differentiate a constant term, it will be equal to zero. The formula for chain rule is given by,
$\dfrac{d}{{dx}}\left[ {f\left( {g\left( x \right)} \right)} \right] = \dfrac{{df\left( {g\left( x \right)} \right)}}{{dx}}{\dfrac{{dg\left( x \right)}}{{dx}}^{^{}}} \\\ \dfrac{{d\left( {\ln \left( {{x^2} + 1} \right)} \right)}}{{dx}} = \dfrac{1}{{{x^2} + 1}}\left( {2x + 0} \right) = \dfrac{{2x}}{{{x^2} + 1}} \;$
This is our required solution.
So, the correct answer is “ $\dfrac{{2x}}{{{x^2} + 1}}$ ”.

Note : Here in this problem, it has a natural log which is represented by $\ln$ . The difference between $\ln$ and $\log$ is, in natural $\ln$ it has the base $e$ in it, while in the log, it has the base $10$ . ${\log _{10}}$ tells you that what power does $10$ has to be raised to get a number $x$ and ${\ln _e}$ , tells us that what power does $e$ has to be raised to get a number $x$ .
This chain rule is applied, when there is function of function in the given equation. It should be remembered forever in our lifetime, because it is important to know this formula to solve many complex equations. $\ln ({x^2} + 1)$ is mentioned as function of function, because $\ln$ is a function and $({x^2} + 1)$ is also another function and hence we represent it as function of function. And for this case we use chain rule to solve this function.