Question

What is the derivative of this function $y = {\sin ^{ - 1}}(2x)$ ?

Answer 1

Hint : We will use the derivative of the inverse function formula to find the derivative in the first step then later we will also use the chain rule of the composite function. Then we will apply the derivative of the given power in the function and then by using the chain rule will take the derivative of the function and then the derivative of the angle.

Complete step-by-step answer :
Take the given expression: $y = {\sin ^{ - 1}}(2x)$
First of all the derivative of cosine function is negative sine function and then apply chain rule multiple times.
$\dfrac{d}{{dx}}[y] = \dfrac{d}{{dx}}[{\sin ^{ - 1}}(2x)]$
Now, apply derivative to the second term in multiple and Using identity - $\dfrac{{d[{{\sin }^{ - 1}}(f(x))]}}{{dx}} = \dfrac{1}{{\sqrt {1 - {{(f(x))}^2}} }}.f'(x)$
By placing formula,
$\dfrac{d}{{dx}}[{\sin ^{ - 1}}(2x)] = \dfrac{1}{{\sqrt {1 - {{(2x)}^2}} }}.\dfrac{d}{{dx}}(2x)$
Now, again apply the derivative –
$\dfrac{d}{{dx}}[{\sin ^{ - 1}}(2x)] = \dfrac{1}{{\sqrt {1 - 4{x^2}} }}.2\dfrac{d}{{dx}}(x)$
Simplify,
$\dfrac{d}{{dx}}[{\sin ^{ - 1}}(2x)] = \dfrac{2}{{\sqrt {1 - 4{x^2}} }}$
This is the required solution.
So, the correct answer is “ $\dfrac{2}{{\sqrt {1 - 4{x^2}} }}$ ”.

Note : Know the difference between the differentiation and the integration and apply formula accordingly. Differentiation can be represented as the rate of change of the function, whereas integration represents the sum of the function over the range. They are inverses of each other.