Question:
If $u=\sec^{-1}(-\sec 2\theta)$ and $v=\cos \theta$, then $\frac{du}{dv}$ at $\theta=\frac{\pi}{4}$ is equal to:
If $u=\sec^{-1}(-\sec 2\theta)$ and $v=\cos \theta$, then $\frac{du}{dv}$ at $\theta=\frac{\pi}{4}$ is equal to:
Show Hint
Simplify the inverse trigonometric expression using identities before differentiating to save time.
Updated On: May 10, 2026
- $\sqrt{2}$
- $2\sqrt{2}$
- $\frac{1}{\sqrt{2}}$
- $\frac{1}{2\sqrt{2}}$
- $-\sqrt{2}$
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The Correct Option is B
Solution and Explanation
Step 1: Concept
Use the derivative of a function with respect to another function: $\frac{du}{dv} = \frac{du/d\theta}{dv/d\theta}$.
Step 2: Analysis
$u = \sec^{-1}(-\sec 2\theta) = \pi - 2\theta$ (using property $\sec^{-1}(-x) = \pi - \sec^{-1}x$). $du/d\theta = -2$. $v = \cos \theta \implies dv/d\theta = -\sin \theta$.
Step 3: Calculation
$\frac{du}{dv} = \frac{-2}{-\sin \theta} = \frac{2}{\sin \theta}$. At $\theta = \pi/4$: $\sin(\pi/4) = 1/\sqrt{2}$. $\frac{du}{dv} = \frac{2}{1/\sqrt{2}} = 2\sqrt{2}$. Final Answer: (B)
Use the derivative of a function with respect to another function: $\frac{du}{dv} = \frac{du/d\theta}{dv/d\theta}$.
Step 2: Analysis
$u = \sec^{-1}(-\sec 2\theta) = \pi - 2\theta$ (using property $\sec^{-1}(-x) = \pi - \sec^{-1}x$). $du/d\theta = -2$. $v = \cos \theta \implies dv/d\theta = -\sin \theta$.
Step 3: Calculation
$\frac{du}{dv} = \frac{-2}{-\sin \theta} = \frac{2}{\sin \theta}$. At $\theta = \pi/4$: $\sin(\pi/4) = 1/\sqrt{2}$. $\frac{du}{dv} = \frac{2}{1/\sqrt{2}} = 2\sqrt{2}$. Final Answer: (B)
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