1. What is the quotient rule formula for differentiating $h(x) = \frac{f(x)}{g(x)}$?
- A.$h'(x) = \frac{f'(x)}{g'(x)}$
- B.$h'(x) = \frac{f'(x)g(x) + f(x)g'(x)}{[g(x)]^2}$
- C.$h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$
- D.$h'(x) = \frac{f(x)g'(x) - f'(x)g(x)}{[g(x)]^2}$
View Answer
Answer: $h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$
Recall the quotient rule: The quotient rule states $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$. The numerator derivative comes first in the numerator. Why the correct answer works: Option C correctly places $f'g$ first, subtracts $fg'$, and divides by $g^2$. Why distractors fail: Option A divides the derivatives directly, which is incorrect. Option B uses addition instead of subtraction (mixing it with the product rule). Option D reverses the subtraction order, which yields the negative of the correct numerator.