Answer: f′(g(x))·g′(x).
- A f′(g(x))·g′(x)
- B the product f′(x)·g′(x)
- C the value f(g′(x))
- D the value g′(f(x))
Correct answer: A. f′(g(x))·g′(x)
Explanation: Differentiate the outer function then multiply by the derivative of the inner function.
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.