Answer: 2/sqrt(1-x 2 ).
- A 2/sqrt(1-x<sup>2</sup>)
- B -2/sqrt(1-x<sup>2</sup>)
- C 1/sqrt(1-x<sup>2</sup>)
- D 2/(1-x<sup>2</sup>)
Correct answer: A. 2/sqrt(1-x<sup>2</sup>)
Explanation: Substituting x = sin(theta), 2x sqrt(1-x<sup>2</sup>) = sin(2theta), so y = sin-1[sin(2theta)] = 2theta = 2sin-1(x) in this range, giving dy/dx = 2/sqrt(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.