Question

Differential coefficient of ${\sin }^{-1}\left(x\right)$ w.r.t.${\cos }^{-1}\left(\sqrt{\left(1-x^2\right)}\right)$ is

• A. $1$
• B. $2$
• C. $\frac{1}{\left(1+x^2\right)}$
• D. none of these

Answer 1

The correct option is A

$1$

Explanation for correct option:

Step-1: Differentiate the ${\sin }^{-1}\left(x\right)$ with respect to $x$.

Let the given, $u={\sin }^{-1}\left(x\right)$

$\Rightarrow \frac{du}{dx}=\frac{1}{\sqrt{1-x^2}}..........\left(i\right)$

Step-2: Differentiate the ${\cos }^{-1}\left(\sqrt{\left(1-x^2\right)}\right)$ with respect to $x$.

Let the given, $v={\cos }^{-1}\left(\sqrt{\left(1-x^2\right)}\right)$

put, $$\begin{gathered} x=\sin \left(\theta \right) \\ \theta ={\sin }^{-1}\left(x\right) \end{gathered}$$

$$\begin{gathered} \Rightarrow v={\cos }^{-1}\left(\sqrt{\left(1-{\left(\sin \left(\theta \right)\right)}^2\right)}\right) \\ \Rightarrow v={\cos }^{-1}\left(\cos \left(\theta \right)\right) \\ \Rightarrow v=\theta \\ \Rightarrow v={\sin }^{-1}\left(x\right) \end{gathered}$$

Differentiate with respect to $x$.

$\Rightarrow \frac{dv}{dx}=\frac{1}{\sqrt{1-{\left(x\right)}^2}}...........\left(ii\right)$

Step-3: Divide equation $\left(i\right)$ by equation $\left(ii\right)$.

$$\begin{gathered} \Rightarrow \frac{\left({\displaystyle \frac{du}{dx}}\right)}{\left({\displaystyle \frac{dv}{dx}}\right)}=\frac{{\displaystyle \frac{1}{\sqrt{\left(1-x^2\right)}}}}{{\displaystyle \frac{1}{\sqrt{\left(1-x^2\right)}}}} \\ \Rightarrow \frac{du}{dv}=1 \end{gathered}$$

Hence, correct answer is option $\left(A\right)$.