1. 3.2 – The Product and Quotient Rules
Math 1304 Calculus I
3.2 – The Product and Quotient Rules

2. 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)

3. 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 x2 = f(0) g(0) + (f(0) g’(0) + f’(0)g(0)) x + b d x2
Find derivative of F(x) = f(x) g(x) at x = 0.

4. 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 =

5. Examples
y = (x+1)/(x-1)

6. 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

7. Proof of Quotient Rule
See book

8. A good working set of rules
Constants: If f(x) = c, then f’(x) = 0
Powers: If f(x) = xn, then f’(x) = nxn-1
Exponentials: If f(x) = ax, then f’(x) = (ln a) ax
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