Question

\(\frac{d}{dx} \left[ \log x^2 + \log a^2 \right] \)

• A. \( \frac{1}{x^2} + \frac{1}{a^2} \)
• B. \( \frac{2}{x} + \frac{2}{a} \)
• C. \( \frac{1}{x} \)
• D. \( \frac{2}{x} \)

Hint:

For the logarithmic function \( \log a^2 \), apply the property \( \log a^2 = 2 \log a \), and then differentiate.

Answer 1

The Correct Option is D

The given function is: \[ f(x) = \log x^2 + \log a^2 \] Step 1: Apply the logarithmic property
We can simplify the expression using the logarithmic identity \( \log a^2 = 2 \log a \), so the expression becomes: \[ f(x) = 2 \log x + 2 \log a \]
Step 2: Differentiate
Now differentiate each term: \[ \frac{d}{dx} \left( 2 \log x \right) = \frac{2}{x} \] The derivative of \( 2 \log a \) with respect to \( x \) is 0 because \( a \) is a constant. Thus, the derivative is: \[ \frac{d}{dx} \left[ 2 \log x + 2 \log a \right] = \frac{2}{x} \]