Answer: 2x cos(x 2 +1).
- A cos(x<sup>2</sup>+1)
- B 2x cos(x<sup>2</sup>+1)
- C 2x sin(x<sup>2</sup>+1)
- D cos(2x)
Correct answer: B. 2x cos(x<sup>2</sup>+1)
Explanation: Let u = x<sup>2</sup>+1. dy/dx = cos(u) times du/dx = cos(x<sup>2</sup>+1) times 2x = 2x cos(x<sup>2</sup>+1).
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.