Answer: 2x/(1+x 4 ).
- A 2x/(1+x<sup>4</sup>)
- B 1/(1+x<sup>4</sup>)
- C 2x/(1+x<sup>2</sup>)
- D x<sup>2</sup>/(1+x<sup>4</sup>)
Correct answer: A. 2x/(1+x<sup>4</sup>)
Explanation: By chain rule with u = x<sup>2</sup>: dy/dx = 1/(1+u<sup>2</sup>) times du/dx = 1/(1+x<sup>4</sup>) times 2x = 2x/(1+x<sup>4</sup>).
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.