Question

If y = cos (sin x²), then at x = $\sqrt{\frac{\pi}{2}}$, $\frac{dy}{dx}$ =

• A.
  • 2
• B. 2
• C. -2$\sqrt{\frac{\pi}{2}}$
• D. 0

Answer 1

Answer: (D)

0

Answer 2

Given:

$$y = \cos (\sin (x^2))$$

Differentiate using the chain rule:

$$\frac{dy}{dx} = -\sin (\sin x^2) \cdot \cos(x^2) \cdot \frac{d}{dx}(x^2)$$

Since $\frac{d}{dx}(x^2) = 2x$, we have:

$$\frac{dy}{dx} = -2x \cos(x^2) \sin (\sin x^2)$$

At $x = \sqrt{\frac{\pi}{2}}$,

$$x^2 = \frac{\pi}{2} \quad \text{and} \quad \cos\left(\frac{\pi}{2}\right)=0.$$

Thus,

$$\frac{dy}{dx} = -2\sqrt{\frac{\pi}{2}} \cdot 0 \cdot \sin\left(\sin\frac{\pi}{2}\right) = 0.$$