Answer: xˣ(1 + ln x).
- A xˣ(1 + ln x)
- B simply x·xˣ⁻¹
- C simply just xˣ
- D simply ln x
Correct answer: A. xˣ(1 + ln x)
Explanation: Using logarithmic differentiation, dy/dx = xˣ(1 + ln 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.