Answer: 1/(2(1+x2)).
- A 1/(2(1+x2))
- B 1/(1+x2)
- C 2/(1+x2)
- D -1/(2(1+x2))
Correct answer: A. 1/(2(1+x2))
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+x2) = 1/(2(1+x2)).
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.