Answer: 2ex cos x.
- A 2ex cos x
- B 2ex(cos x - sin x)
- C 2ex sin x
- D ex(cos x - sin x)
Correct answer: A. 2ex cos x
Explanation: Product rule: dy/dx = eˣ sin x + eˣ cos x = eˣ(sin x + cos x). Differentiate again: d²y/dx² = eˣ(sin x + cos x) + eˣ(cos x − sin x) = eˣ(2cos x) = 2eˣ cos 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.