Question

If $y = a \cos(\log x) + b \sin(\log x)$, then $x^2y'' + xy'1$ is:

• A. $\cot(\log x)$
• B. $y$
• C. $-y$
• D. $\tan(\log x)$

Hint:

When differentiating trigonometric functions involving logarithms, remember to apply the chain rule for the logarithmic argument.

Answer 1

The Correct Option is B

We are given the function $y = a \cos(\log x) + b \sin(\log x)$. Let's compute the first and second derivatives of $y$. First derivative: \[ y' = a \frac{d}{dx}[\cos(\log x)] + b \frac{d}{dx}[\sin(\log x)] = -a \frac{1}{x} \sin(\log x) + b \frac{1}{x} \cos(\log x). \] Second derivative: \[ y" = -a \frac{d}{dx}\left(\frac{1}{x} \sin(\log x)\right) + b \frac{d}{dx}\left(\frac{1}{x} \cos(\log x)\right). \] After simplification, we get: \[ x^2y" + xy'1 = y. \]