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# The Product and Quotient Rules

### The Product Rule

 Theorem  (The Product Rule) Let f and g be differentiable functions.  Then $\left[f(x) \, g(x)\right] ' = f(x)\, g '(x) + f '(x) \,g(x)$

Proof:

We have

Example $$\PageIndex{1}$$:

Find

d
(2 - x2)(x4 - 5)
dx

Solution:

Here

f(x)  =  2 - x2

and

g(x)  =  x4 - 5

The product rule gives

d
(2 - x2)(x4 - 5)  =  (2 - x2)(4x3) + (-2x)(x4 - 5)
dx

### The Quotient Rule

Remember the poem

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

This poem is the mnemonic for the taking the derivative of a quotient.

 Theorem $\dfrac{d}{dx} \dfrac{f(x)}{g(x)} = \dfrac{g(x)\, f'(x) - f(x) \, g'(x)}{g(x)^2}$

Example $$\PageIndex{2}$$:

Find y' if

2x - 1
y'  =
x + 1

Solution:

Here

f(x) = 2x - 1

and

g(x) = x + 1

The quotient rule gives

(x + 1)(2) - (2x - 1)(1)

(x + 1)2

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

3
=
(x + 1)2