Answer: xsin x [cos x ln x + (sin x)/x].
- A xsin x [cos x ln x + (sin x)/x]
- B xsin x cos x ln x, missing the second additive term
- C xsin x (sin x)/x, missing the logarithmic term
- D sin x times xsin x - 1, a plain power-rule shortcut
Correct answer: A. xsin x [cos x ln x + (sin x)/x]
Explanation: Take ln: ln y = sin x times ln x. Differentiate: (1/y)(dy/dx) = cos x ln x + sin x times (1/x). So dy/dx = y[cos x ln x + (sin x)/x] = xsin x[cos x ln x + (sin x)/x].
f(x)=|x| is perfectly continuous at x=0 (no break, no jump), but the curve has a sharp corner there: approaching from the left gives slope -1 and from the right gives slope +1 - since these one-sided derivatives disagree, the function is not differentiable at that point.
Concept context
When a function has no breaks (continuity) and when it has a well-defined slope (differentiability), plus rules for differentiating implicit, inverse trig, exponential, log, and parametric functions.