Answer: xsinx(cosx·lnx + sinx/x).
- A xsinx(cosx·lnx + sinx/x)
- B xsinx·cosx
- C sinx·xsinx − 1
- D xsinx·lnx
Correct answer: A. xsinx(cosx·lnx + sinx/x)
Explanation: Logarithmic differentiation gives dy/dx = xsinx(cosx·lnx + sinx/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.