Answer: 1/(2(1+x 2 )).
- A 1/(2(1+x<sup>2</sup>))
- B 1/(1+x<sup>2</sup>)
- C 2/(1+x<sup>2</sup>)
- D -1/(2(1+x<sup>2</sup>))
Correct answer: A. 1/(2(1+x<sup>2</sup>))
Explanation: Substituting x = tan(theta), the expression simplifies to tan-1[tan(theta/2)] = theta/2 = (1/2)tan-1(x), so dy/dx = (1/2) times 1/(1+x<sup>2</sup>) = 1/(2(1+x<sup>2</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.