Question

The derivative of \( \sin x \) with respect to \( \log x \) is:

• A. \( x \cos x \)
• B. \( \cos x \log x \)
• C. \( \cos x \)
• D. \( \cos x \)

Hint:

When differentiating with respect to a logarithmic function, use the chain rule by first differentiating the numerator and denominator, then multiplying them together.

Answer 1

The Correct Option is A

Let \( y = \sin x \), and \( z = \log x \). To differentiate with respect to \( \log x \), we use the chain rule: \[ \frac{dy}{dz} = \frac{dy}{dx} \times \frac{dx}{dz} \] Now, compute each term: \[ \frac{dy}{dx} = \cos x \quad \text{(derivative of \( \sin x \))} \] \[ \frac{dx}{dz} = \frac{1}{x} \quad \text{(derivative of \( \log x \))} \] So, \[ \frac{dy}{dz} = \cos x \times \frac{1}{x} = \frac{\cos x}{x} \] Thus, the correct answer is option (1) \( x \cos x \).