Math 1304 Calculus I 3.2 – The Product and Quotient Rules

The Product Rule Statement of the product rule: If F = f g is the product of two functions, then F'(x) = f’(x)g(x)+f(x)g’(x)

Exploration Consider the product F(x) of two linear functions: f(x) = a + b x, g(x) = c + d x f(0) = a, f’(0) = b, g(0) = c, g’(0) = d Then f(x) = f(0) + f’(0) x, g(x) = g(0) + g’(0)x Then f(x) g(x) = (a + b x)(c + d x) = a c + (a d + b c) x + b d x 2 = f(0) g(0) + (f(0) g’(0) + f’(0)g(0)) x + b d x 2 Find derivative of F(x) = f(x) g(x) at x = 0.

Proof of Product Rule F(x) = f(x) g(x) F(a+h) = F(a) = F(a+h) – F(a) = (F(a+h) – F(a))/h =

Examples y = (x+1)(x-1)

Ways of Stating The Quotient Rule Statement of the quotient rule: If F = f /g is the quotient of two functions, then F'(x) =( f’(x)g(x) - f(x)g’(x))/(g’(x)) 2

Proof of Quotient Rule See book

A good working set of rules Constants: If f(x) = c, then f’(x) = 0 Powers: If f(x) = x n, then f’(x) = nx n-1 Exponentials: If f(x) = a x, then f’(x) = (ln a) a x Scalar mult: If f(x) = c g(x), then f’(x) = c g’(x) Sum: If f(x) = g(x) + h(x), then f’(x) = g’(x) + h’(x) Difference: If f(x) = g(x) - h(x), then f’(x) = g’(x) - h’(x) Multiple sums: derivative of sum is sum of derivatives Linear combinations: derivative of linear combo is linear combo of derivatives Product: If f(x) = g(x) h(x), then f’(x) = g’(x) h(x) + g(x)h’(x) Multiple products: If F(x) = g(x) h(x) k(x), then F’(x) = g’(x) h(x) k(x) + g(x) h’(x) k(x) + g(x) h(x) k’(x) Quotient: If f(x) = g(x)/h(x), then f’(x) = (g’(x) h(x) - g(x)h’(x))/(h(x)) 2