Question

Find \( \frac{d}{dx} \left( \log 5x \right) \).

• A. \( \frac{1}{5x} \)
• B. \( \frac{1}{x} \)
• C. \( \frac{5}{x} \)
• D. \( \log 5 + \frac{1}{x} \)

Hint:

When differentiating logarithmic functions, use the chain rule to account for any coefficients inside the logarithm.

Answer 1

The Correct Option is A

We are asked to differentiate \( \log 5x \) with respect to \( x \). Using the chain rule: \[ \frac{d}{dx} \left( \log 5x \right) = \frac{d}{dx} \left( \log 5 + \log x \right) \] Since \( \log 5 \) is a constant, its derivative is zero. Thus: \[ \frac{d}{dx} \left( \log 5x \right) = \frac{d}{dx} \left( \log x \right) = \frac{1}{x} \] But because we have \( 5x \) and not just \( x \), we must apply the chain rule: \[ \frac{d}{dx} \left( \log 5x \right) = \frac{1}{5x} \times \frac{d}{dx} (5x) = \frac{1}{5x} \times 5 = \frac{1}{x} \] Thus, the correct answer is \( \frac{1}{5x} \).