Answer: y[x cot x + ln(sin x)].
- A y[x cot x + ln(sin x)]
- B y[x cot x]
- C y[ln(sin x)]
- D x(sin x)<sup>x-1</sup> cos x
Correct answer: A. y[x cot x + ln(sin x)]
Explanation: Take ln: ln y = x ln(sin x). Differentiating: (1/y)(dy/dx) = ln(sin x) + x cot x. So dy/dx = y[x cot x + ln(sin 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.