Question

How do you find the derivative of $y = {\left( {\ln x} \right)^2}$ ?

Answer 1

Hint: As we observe that the equation contains composite functions, in which we will differentiate outer function then inner function, as in above composite function we will differentiate power function as it is outermost and then we will differentiate $\ln x$.

Complete step by step solution:
To find out the derivative of $y = {\left( {\ln x} \right)^2}$ we should know that we will perform chain rule of differentiation. As it contains two functions square and $\ln x$.
$y = {\left( {\ln x} \right)^2}$
We will perform differentiation on both sides of the equation.
First of all, we differentiate the power function in which the power of a variable will become coefficient and the power of the variable will be subtracted by one.
And then perform differentiate of $\ln x$ which is equals $\dfrac{1}{x}$
$\dfrac{{dy}}{{dx}} = 2(\ln x) \times \dfrac{1}{x}$
$ \Rightarrow \dfrac{{2(\ln x)}}{x}$

Hence, the derivative of $y$ is $\dfrac{{2(\ln x)}}{x}$.

Note:
We should recognize the composite function in the question. Also, we should know that we will apply the chain differentiation rule to get the solution in the composite function. In this rule, we differentiate outer function and then inner function.