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: Differentiate x = cos y
$1=-\sin y\,\frac{dy}{dx}$, so $\frac{dy}{dx}=-\csc y$.

Step 2: Differentiate again
Using the chain rule, $\frac{d^{2}y}{dx^{2}}=\frac{d}{dy}(-\csc y)\cdot\frac{dy}{dx}$.

Step 3: Substitute
$\frac{d}{dy}(-\csc y)=\csc y\cot y$, so the result is $\csc y\cot y\cdot(-\csc y)=-\csc^{2}y\cot y$.

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