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Question: How do you find the derivative of \({{\left( \arcsin x \right)}^{2}}\)?...

How do you find the derivative of (arcsin⁡x)2{{\left( \arcsin x \right)}^{2}}?

Explanation

Solution

In this question we will find the derivative of the given trigonometric function by using the chain rule since it is in the form of a composite function. It has two functions in it which is the arcsin⁡x\arcsin xfunction, which is also called the inverse sine function. And we have the square function. We will use the formula of derivative that ddxxn=nxn−1\dfrac{d}{dx}{{x}^{n}}=n{{x}^{n-1}}and ddx(arcsin⁡x)=11−x2\dfrac{d}{dx}(\arcsin x)=\dfrac{1}{\sqrt{1-{{x}^{2}}}}to substitute in the expression and simplify to get the required solution.

Complete step-by-step solution:
We have the expression given to us as (arcsin⁡x)2{{\left( \arcsin x \right)}^{2}}
Since we have to take the derivative of the term, we can write it as:
⇒ddx(arcsin⁡x)2\Rightarrow \dfrac{d}{dx}{{\left( \arcsin x \right)}^{2}}
Now since there is no direct formula for calculating the derivative of the given expression, we will use the chain rule. The formula for the chain rule is: F′(x)=f′(g(x))g′(x)F'(x)=f'(g(x))g'(x).
Now in this question we have a composite function in the form of f(g(x))f(g(x))where f(x)=x2f(x)={{x}^{2}} and g(x)=arcsin⁡xg(x)=\arcsin x.
Now we know that ddxxn=nxn−1\dfrac{d}{dx}{{x}^{n}}=n{{x}^{n-1}} and since we have to use the chain rule on the expression, we can write it as:
⇒2(arcsin⁡x)×ddx(arcsin⁡x)\Rightarrow 2(\arcsin x)\times \dfrac{d}{dx}(\arcsin x)
Now we know that ddx(arcsin⁡x)=11−x2\dfrac{d}{dx}(\arcsin x)=\dfrac{1}{\sqrt{1-{{x}^{2}}}} therefore, on using the formula and substituting, we get:
⇒2(arcsin⁡x)×11−x2\Rightarrow 2(\arcsin x)\times \dfrac{1}{\sqrt{1-{{x}^{2}}}}
On simplifying the terms by multiplying, we get:
⇒2(arcsin⁡x)1−x2\Rightarrow \dfrac{2(\arcsin x)}{\sqrt{1-{{x}^{2}}}}, which is the required solution.

Note: In this question we have the inverse trigonometric function arcsin⁡x\arcsin x which is called the inverse sine function or the sine inverse function. It is also represented as sin⁡−1x{{\sin }^{-1}}x.
The chain rule is to be used when there are multiple functions in the expression. In this question there are 22 functions, there can be more than 22 functions in an expression too.
The inverse of the derivative is the integration and vice versa. If the derivative of a term aa is bb, then the integration of the term bb will be aa.