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Question: If \(y={{\left( {{\sin }^{-1}}x \right)}^{2}}\), then \(\left( 1-{{x}^{2}} \right)\dfrac{{{d}^{2}}y}...

If y=(sin⁡−1x)2y={{\left( {{\sin }^{-1}}x \right)}^{2}}, then (1−x2)d2ydx2−xdydx\left( 1-{{x}^{2}} \right)\dfrac{{{d}^{2}}y}{d{{x}^{2}}}-x\dfrac{dy}{dx} is equal to
A. 1
B. 0
C. −1-1
D. 2

Explanation

Solution

We first define the chain rule and how the differentiation of composite function works. We differentiate the main function with respect to the intermediate function and then take differentiation of the intermediate function with respect to xx. We take multiplication of these two different differentiated values.

Complete step by step answer:
We differentiate the given function f(x)=(sin⁡−1x)2f\left( x \right)={{\left( {{\sin }^{-1}}x \right)}^{2}} with respect to xx using the chain rule.
Here we have a composite function where the main function is g(x)=x2g\left( x \right)={{x}^{2}} and the other function is h(x)=sin⁡−1xh\left( x \right)={{\sin }^{-1}}x.
We have goh(x)=g(sin⁡−1x)=(sin⁡−1x)2goh\left( x \right)=g\left( {{\sin }^{-1}}x \right)={{\left( {{\sin }^{-1}}x \right)}^{2}}. We take this as ours f(x)=(sin⁡−1x)2f\left( x \right)={{\left( {{\sin }^{-1}}x \right)}^{2}}.
We need to find the value of ddx[f(x)]=ddx[(sin⁡−1x)2]\dfrac{d}{dx}\left[ f\left( x \right) \right]=\dfrac{d}{dx}\left[ {{\left( {{\sin }^{-1}}x \right)}^{2}} \right]. We know f(x)=goh(x)f\left( x \right)=goh\left( x \right).
Differentiating f(x)=goh(x)f\left( x \right)=goh\left( x \right), we get
ddx[f(x)]=ddx[goh(x)]=dd[h(x)][goh(x)]×d[h(x)]dx=g′[h(x)]h′(x)\dfrac{d}{dx}\left[ f\left( x \right) \right]=\dfrac{d}{dx}\left[ goh\left( x \right) \right]=\dfrac{d}{d\left[ h\left( x \right) \right]}\left[ goh\left( x \right) \right]\times \dfrac{d\left[ h\left( x \right) \right]}{dx}={{g}^{'}}\left[ h\left( x \right) \right]{{h}^{'}}\left( x \right).
The above-mentioned rule is the chain rule.
The chain rule allows to differentiate with respect to the function h(x)h\left( x \right) instead of xx and after that we need to take the differentiated form of h(x)h\left( x \right) with respect to xx.
For the function f(x)=(sin⁡−1x)2f\left( x \right)={{\left( {{\sin }^{-1}}x \right)}^{2}}, we take differentiation of f(x)=(sin⁡−1x)2f\left( x \right)={{\left( {{\sin }^{-1}}x \right)}^{2}} with respect to the function h(x)=sin⁡−1xh\left( x \right)={{\sin }^{-1}}x instead of xx and after that we need to take the differentiated form of h(x)=sin⁡−1xh\left( x \right)={{\sin }^{-1}}x with respect to xx.
We know that differentiation of g(x)=x2g\left( x \right)={{x}^{2}} is g′(x)=2x{{g}^{'}}\left( x \right)=2x and differentiation of h(x)=sin⁡−1xh\left( x \right)={{\sin }^{-1}}x is h′(x)=11−x2{{h}^{'}}\left( x \right)=\dfrac{1}{\sqrt{1-{{x}^{2}}}}. We apply the formula of ddx(xn)=nxn−1\dfrac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}}.
⇒ddx[f(x)]=dd[sin⁡−1x][(sin⁡−1x)2]×d[sin⁡−1x]dx\Rightarrow \dfrac{d}{dx}\left[ f\left( x \right) \right]=\dfrac{d}{d\left[ {{\sin }^{-1}}x \right]}\left[ {{\left( {{\sin }^{-1}}x \right)}^{2}} \right]\times \dfrac{d\left[ {{\sin }^{-1}}x \right]}{dx}
We place the values of the differentiations and get
⇒ddx[f(x)]=2sin⁡−1x[11−x2]=2sin⁡−1x1−x2\Rightarrow \dfrac{d}{dx}\left[ f\left( x \right) \right]=2{{\sin }^{-1}}x\left[ \dfrac{1}{\sqrt{1-{{x}^{2}}}} \right]=\dfrac{2{{\sin }^{-1}}x}{\sqrt{1-{{x}^{2}}}}
We take square of dydx=2sin⁡−1x1−x2\dfrac{dy}{dx}=\dfrac{2{{\sin }^{-1}}x}{\sqrt{1-{{x}^{2}}}} and get (dydx)2=4(sin⁡−1x)21−x2{{\left( \dfrac{dy}{dx} \right)}^{2}}=\dfrac{4{{\left( {{\sin }^{-1}}x \right)}^{2}}}{1-{{x}^{2}}}.
Now replacing the values, we get (1−x2)(dydx)2=4(sin⁡−1x)2=4y\left( 1-{{x}^{2}} \right){{\left( \dfrac{dy}{dx} \right)}^{2}}=4{{\left( {{\sin }^{-1}}x \right)}^{2}}=4y.
We differentiate again to get

& \left( 1-{{x}^{2}} \right){{\left( \dfrac{dy}{dx} \right)}^{2}}=4{{\left( {{\sin }^{-1}}x \right)}^{2}}=4y \\\ & \Rightarrow \left( 1-{{x}^{2}} \right)\times 2\left( \dfrac{dy}{dx} \right)\dfrac{{{d}^{2}}y}{d{{x}^{2}}}-2x{{\left( \dfrac{dy}{dx} \right)}^{2}}=4\dfrac{dy}{dx} \\\ & \Rightarrow \left( 1-{{x}^{2}} \right)\dfrac{{{d}^{2}}y}{d{{x}^{2}}}-x\dfrac{dy}{dx}=2 \\\ \end{aligned}$$ **Therefore, $\left( 1-{{x}^{2}} \right)\dfrac{{{d}^{2}}y}{d{{x}^{2}}}-x\dfrac{dy}{dx}$ is equal to 2.** **Note:** We need remember that in the chain rule $$\dfrac{d}{d\left[ h\left( x \right) \right]}\left[ goh\left( x \right) \right]\times \dfrac{d\left[ h\left( x \right) \right]}{dx}$$, we aren’t cancelling out the part $$d\left[ h\left( x \right) \right]$$. Cancellation of the base differentiation is never possible. It’s just a notation to understand the function which is used as a base to differentiate.