Answer: dy/dx = xx (1 + ln x).
- A dy/dx = xx
- B dy/dx = xx (1 + ln x)
- C dy/dx = x times xx-1
- D dy/dx = xx ln x
Correct answer: B. dy/dx = xx (1 + ln x)
Explanation: Take ln: ln y = x ln x. Differentiate: (1/y)(dy/dx) = ln x + 1. So dy/dx = y(1 + ln x) = 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.