The Product and Quotient Rules

I.  Quiz

II.  Homework

III.  The Product Rule

Theorem:

Let f and g be differentiable functions.  Then

(f(x)g(x))' = f(x)g'(x) + f'(x)g(x)

Proof:

We have d/dx (fg) = lim [f(x+h)g(x+h) - f(x)g(x)]/h

= lim [f(x+h)g(x+h)  - f(x+h)g(x) + f(x+h)g(x) - f(x)g(x)]/h

= lim [f(x+h)(g(x+h) - g(x)/h + (g(x)(f(x+h) - f(x))/h]

= [lim f(x+h)]g'(x) + g(x)f'(x)

= f(x)g'(x) + g(x)f'(x)

Example

d/dx[(2 - x²)(x⁴ - 5)]

Solution:    (2 - x²)(4x³) + (-2x)(x⁴ - 5)

Other examples will be given.

IV  The Quotient Rule

Remember the poem

"lo d hi minus hi d lo square the bottom and away you go"

Theorem:

d/dx(f/g) = [gf' - fg']/g²  

Example:

find y' if

y = (2x - 1)/(x + 1)

Solution:

[(x + 1)(2) - (2x - 1)(1)]/(x + 1)²  = 2/(x + 1)²  

Exercises will be given.