Answer: dy/dx = x x (1 + ln x).
- A dy/dx = x<sup>x</sup>
- B dy/dx = x<sup>x</sup> (1 + ln x)
- C dy/dx = x times x<sup>x-1</sup>
- D dy/dx = x<sup>x</sup> ln x
Correct answer: B. dy/dx = x<sup>x</sup> (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) = x<sup>x</sup>(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.