Answer: x = tan(theta), since 2x/(1-x 2 ) is tan(2theta) when x = tan(theta).
- A x = sin(theta), reducing the expression to a sine double-angle form
- B x = tan(theta), since 2x/(1-x<sup>2</sup>) is tan(2theta) when x = tan(theta)
- C x = cos(theta), reducing the expression to a cosine double-angle form
- D x = sec(theta), used for expressions involving square roots of x²-1
Correct answer: B. x = tan(theta), since 2x/(1-x<sup>2</sup>) is tan(2theta) when x = tan(theta)
Explanation: Using x = tan(theta), 2x/(1-x<sup>2</sup>) = tan(2theta), so y = 2theta = 2tan-1(x), giving dy/dx = 2/(1+x<sup>2</sup>) directly.
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.