Which expression is the correct quotient rule for d/dx [f(x)/g(x)]?

Which expression is the correct quotient rule for d/dx [f(x)/g(x)]?

• A. [f'(x) g(x) - f(x) g'(x)] / [g(x)]^2
• B. [f(x) g'(x) - g(x) f'(x)] / [g(x)]^2
• C. [f'(x) g(x) + f(x) g'(x)] / [g(x)]^2
• D. [g(x) f'(x) - f(x) g'(x)] / [g(x)]^2

Answer: (D)

When differentiating a quotient, use the quotient rule: the derivative of f(x)/g(x) is (f'(x) g(x) − f(x) g'(x)) / [g(x)]^2. The minus sign comes from differentiating the reciprocal 1/g(x) via the chain rule: d/dx (1/g) = −g'/g^2. Writing f/g as f · (1/g) and applying the product rule gives f'·(1/g) + f·(−g'/g^2) = (f' g − f g') / g^2. This matches the option with g f' − f g' in the numerator over g^2, since multiplication is commutative. So that form is correct. The plus form would not arise from the quotient rule, and swapping the order inside the subtraction changes the sign, which would not give the derivative.

When differentiating a quotient, use the quotient rule: the derivative of f(x)/g(x) is (f'(x) g(x) − f(x) g'(x)) / [g(x)]^2. The minus sign comes from differentiating the reciprocal 1/g(x) via the chain rule: d/dx (1/g) = −g'/g^2. Writing f/g as f · (1/g) and applying the product rule gives f'·(1/g) + f·(−g'/g^2) = (f' g − f g') / g^2. This matches the option with g f' − f g' in the numerator over g^2, since multiplication is commutative. So that form is correct. The plus form would not arise from the quotient rule, and swapping the order inside the subtraction changes the sign, which would not give the derivative.