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