If \(y = \cos\sqrt{x}\), \(x \gt 0\), then \(\displaystyle\frac{dy}{dx}\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(-\displaystyle\frac{\sin\sqrt{x}}{2\sqrt{x}}\)
Let \(u = \sqrt{x}\), so \(y = \cos u\). By the chain rule:
\( \displaystyle \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \)
\( \displaystyle = (-\sin u) \cdot \frac{1}{2\sqrt{x}} \)
\( \displaystyle = -\frac{\sin\sqrt{x}}{2\sqrt{x}} \)