AP EAMCET · Maths · Differentiation
Which one of the following is false?
- A \(\frac{d}{d x}\left[\sec ^{-1}(\cos h x)\right]=\sec h x\)
- B \(\frac{d}{d x}\left[\cos ^{-1}(\sec h x)\right]=\sec h x\)
- C \(\frac{d}{d x}\left[\tan ^{-1}(\sin h x)\right]=\sec h x\)
- D \(\frac{d}{d x}\left[\tan ^{-1}\left(\tan \frac{x}{2}\right)\right]=\sec x\)
Answer & Solution
Correct Answer
(D) \(\frac{d}{d x}\left[\tan ^{-1}\left(\tan \frac{x}{2}\right)\right]=\sec x\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \text { } \frac{d}{d x}\left[\tan ^{-1}\left(\tan \frac{x}{2}\right)\right]=\frac{1}{\tan ^2 \frac{x}{2}+1} \times \sec ^2 \frac{x}{2} \times \frac{1}{2} \\ & \frac{\sec ^2 \frac{x}{2}}{\sec ^2 \frac{x}{2}} \times \frac{1}{2}=\frac{1}{2}\end{aligned}\)
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