Question

How do you find the derivative of ${\left( {\ln \left( {{x^2} + 3} \right)} \right)^3}$?

Answer 1

We simply use chain rule of differentiation to calculate the derivative of the given function with respect to the variable ‘x’. First differentiate the power of the function, then differentiate the logarithm part of the function and in the end differentiate the innermost bracket.

• General formula of differentiation is$\dfrac{d}{{dx}}({x^n}) = n{x^{n - 1}}$
• Differentiation of a log function is given by $\dfrac{d}{{dx}}(\log x) = \dfrac{1}{x}$
• Chain rule of differentiation:$\dfrac{d}{{dx}}g\left( {f(x)} \right) = \dfrac{d}{{dx}}g(f(x)) \times \dfrac{d}{{dx}}f(x)$

Complete step-by-step answer:
We are given the function ${\left( {\ln \left( {{x^2} + 3} \right)} \right)^3}$
We have to find the derivative of the function with respect to ‘x’. So, we differentiate the function with respect to ‘x’. We use concepts of differentiation to calculate $\dfrac{d}{{dx}}{\left( {\ln \left( {{x^2} + 3} \right)} \right)^3}$.
Here we are given three functions inside the main function.
If we look at the function ${\left( {\ln \left( {{x^2} + 3} \right)} \right)^3}$ we can write
$f(x) = {\left( {g(x)} \right)^3}$; $g(x) = \ln (h(x))$ and $h(x) = {x^2} + 3$
So, using chain rule of differentiation we can solve
$\Rightarrow \dfrac{d}{{dx}}{\left( {\ln \left( {{x^2} + 3} \right)} \right)^3} = \dfrac{{df(x)}}{{dx}}$
Now break the function on right hand side of the equation
$\Rightarrow \dfrac{d}{{dx}}{\left( {\ln \left( {{x^2} + 3} \right)} \right)^3} = \dfrac{{df}}{{dg}} \times \dfrac{{dg}}{{dh}} \times \dfrac{{dh}}{{dx}}$
Now differentiate the functions n right hand side of the equation accordingly.$\Rightarrow \dfrac{d}{{dx}}{\left( {\ln \left( {{x^2} + 3} \right)} \right)^3} = \dfrac{{6x{{\left( {\ln \left( {{x^2} + 3} \right)} \right)}^2}}}{{\left( {{x^2} + 3} \right)}}$
$\Rightarrow \dfrac{d}{{dx}}{\left( {\ln \left( {{x^2} + 3} \right)} \right)^3} = 3{\left( {\ln \left( {{x^2} + 3} \right)} \right)^2} \times \dfrac{1}{{\left( {{x^2} + 3} \right)}} \times 2x$
Multiply the possible products on right hand side of the equation

$\therefore$The derivative of the function ${\left( {\ln \left( {{x^2} + 3} \right)} \right)^3}$is $\dfrac{{6x{{\left( {\ln \left( {{x^2} + 3} \right)} \right)}^2}}}{{\left( {{x^2} + 3} \right)}}$

Note:
Many students make mistakes solving for the derivative of the given function using the log property i.e. $\log {m^n} = n\log m$ which is not at all valid here. Students don’t notice the difference between power of log and power on the complete function that includes logarithm and directly apply the property which is wrong. Keep in mind here we have the complete function as a function with power 3 so we will solve it using the main differentiation formula and then differentiate log with its formula.