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

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

• A. g(x) f'(x) + f(x) g'(x)
• B. f'(x) g(x) - f(x) g'(x)
• C. f(x) g(x)
• D. f'(x) g'(x)

Answer: (A)

Differentiating a product uses the product rule: the rate of change of a product f(x)g(x) is obtained by taking the derivative of one function and multiplying by the other, plus the derivative of the second function times the first. So d/dx [f(x) g(x)] = f'(x) g(x) + f(x) g'(x). Since multiplication is commutative, that can also be written as g(x) f'(x) + f(x) g'(x). The expression with the two terms added matches this rule exactly. The other forms aren’t correct for the product rule: subtracting a term would correspond to a different operation, the original product f(x)g(x) isn’t differentiated, and multiplying the derivatives f'(x)g'(x) isn’t what the product rule prescribes.

Differentiating a product uses the product rule: the rate of change of a product f(x)g(x) is obtained by taking the derivative of one function and multiplying by the other, plus the derivative of the second function times the first. So d/dx [f(x) g(x)] = f'(x) g(x) + f(x) g'(x). Since multiplication is commutative, that can also be written as g(x) f'(x) + f(x) g'(x). The expression with the two terms added matches this rule exactly.

The other forms aren’t correct for the product rule: subtracting a term would correspond to a different operation, the original product f(x)g(x) isn’t differentiated, and multiplying the derivatives f'(x)g'(x) isn’t what the product rule prescribes.