Answer: 2x/(1+x4).
- A 2x/(1+x4)
- B 1/(1+x4)
- C 2x/(1+x2)
- D x2/(1+x4)
Correct answer: A. 2x/(1+x4)
Explanation: By chain rule with u = x2: dy/dx = 1/(1+u2) times du/dx = 1/(1+x4) times 2x = 2x/(1+x4).
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.