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