Question

Find \[\dfrac{d}{{dx}}\left[ {{{\sin }^{ - 1}}\left( {3x - 4{x^3}} \right)} \right]\] equals to

Answer 1

Hint: Given problem is very simple and the hint is given in the problem only. But we will use substitution for the problem to solve. Also we need to know the formula for triple angle also. We will substitute \[x = \sin \theta \] . then the equation given will change and after simplifying this we will find the derivative.

Complete step-by-step answer:
Given that,
\[\dfrac{d}{{dx}}\left[ {{{\sin }^{ - 1}}\left( {3x - 4{x^3}} \right)} \right]\]
We will substitute x to solve this further.
\[x = \sin \theta \]
Now the equation will be,
\[ = {\sin ^{ - 1}}\left( {3\sin \theta - 4{{\sin }^3}\theta } \right)\]
We know that the bracket gives a triple angle for sin function.
\[ = {\sin ^{ - 1}}\left( {\sin 3\theta } \right)\]
We know that, \[{\sin ^{ - 1}}\left( {\sin \theta } \right) = \theta \]
\[ = 3\theta \]
Thus, substitute the value of \[\theta \],
\[ = 3{\sin ^{ - 1}}x\]
Now we will take the derivative,
\[\dfrac{d}{{dx}}3{\sin ^{ - 1}}x\]
Taking the constant outside,
\[ = 3\dfrac{d}{{dx}}{\sin ^{ - 1}}x\]
Taking the derivative now,
\[ = 3.\dfrac{1}{{\sqrt {1 - {x^2}} }}\]
Thus, this is the correct answer.

Note: Note that, don’t start directly with the derivative of the function. That will be very time consuming as well as messy to solve. Always take help of respective trigonometric functions or methods like substitution or elimination for this. Don’t substitute the x in the answer.
We should be masters in handling and shuffling the trigonometric functions. Also note that, all trigonometric identities and formulas should be on the tip of the finger always.