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  • Explain solution RD Sharma class 12 chapter Differentiation exercise 10.8 question 4 sub question (ii) maths

Explain solution RD Sharma class 12 chapter Differentiation exercise 10.8 question 4 sub question (ii) maths

Answers (1)

Answer: -1

Hint: \text { Let } u=\sin ^{-1} \sqrt{1-x^{2}}, v=\cos ^{-1} x

        \frac{d u}{d v}=\frac{\frac{d u}{d x}}{\frac{d v}{d x}}
 

Given:    \sin ^{-1} \sqrt{1-x^{2}} \text { w.r.t } \cos ^{-1} x, x \in(-1,0)

Explanation:

\text { Let } u=\sin ^{-1} \sqrt{1-x^{2}}, v=\cos ^{-1} x

\begin{aligned} &u=\sin ^{-1} \sqrt{1-x^{2}} \\\\ &\text { Let } x=\cos \theta \\\\ &u=\sin ^{-1} \sqrt{1-\cos ^{2} \theta} \end{aligned}

\begin{aligned} &u=\sin ^{-1}(\sin \theta) \\\\ &x \in(-1,0) \\\\ &\cos \theta \in(-1,0) \quad \theta \in\left(-\frac{\pi}{2}, 0\right) \end{aligned}

\begin{aligned} &u=\sin ^{-1}(\sin \theta)=-\theta \quad \theta \in\left(-\frac{\pi}{2}, 0\right) \\\\ &u=-\cos ^{-1} x \end{aligned}

\begin{aligned} &\frac{d u}{d x}=\frac{-1}{\sqrt{1-x^{2}}} \\\\ &v=\cos ^{-1} x \\\\ &\frac{d v}{d x}=\frac{1}{\sqrt{1-x^{2}}} \end{aligned}

\frac{d u}{d v}=\frac{\frac{d u}{d x}}{\frac{d v}{d x}}=\frac{\frac{-1}{\sqrt{1-x^{2}}}}{\frac{1}{\sqrt{1-x^{2}}}}=-1

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