Question

If \(y=\cos ^{-1}x\), then \(\frac{d^{2}y}{dx^{2}}=\)

• A. \(\sec ^{2}y\cdot \tan y\)
• B. \(-\sec ^{2}y\cdot \tan y\)
• C. \(\csc ^{2}y\cdot \cot y\)
• D. \(-\csc ^{2}y\cdot \cot y\)

Hint:

Write y = cos inverse x so x = cos y, differentiate once to get minus cosec y, then differentiate again with respect to x.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
Since \(y=\cos^{-1}x\), we have \(x=\cos y\) and \(\sqrt{1-x^{2}}=\sin y\).

Step 2: First derivative
\(\frac{dy}{dx}=-\frac{1}{\sqrt{1-x^{2}}}=-\frac{1}{\sin y}=-\csc y\).

Step 3: Second derivative
\(\frac{d^{2}y}{dx^{2}}=\csc y\cot y\cdot\frac{dy}{dx}=\csc y\cot y\,(-\csc y)=-\csc^{2}y\cot y\).

Final Answer:
The second derivative is \(-\csc^{2}y\cdot\cot y\). \[ \boxed{\text{(D) }-\csc ^{2}y\cdot \cot y} \]