2.3 The product rule, quotient rule, and higher order derivatives

"Life is too short for long division."

Objective

Derivative RulesConstant RulePower RuleConstant Multiple RuleDerivative of x lemmaTrig RulesSum and Difference Rule

Limit RulesSum and difference of limitsProduct of limitsQuotient of limits

What about…

134 22 xxyFirst Way Second Way

Product RuleThe derivative of the product of 2

differentiable functions f and g is differentiable. Specifically:

In other words, h’(x) = (derivative of the first) times

(second) + (derivative of the second) times (first)

)()(')()(')()( xfxgxgxfxgxfdxd

Example

xxxxh 4523)( 2

Another example

xxy sin

One more example

xxxy sin2cos2

Quotient RuleThe derivative of f/g of 2 differentiable

functions f and g is differentiable at all values as long as g(x) =/ 0. Specifically:

In other words:

2)()(')()(')(

)()(

xgxgxfxfxg

xgxf

dxd

2bottom

bottomdxdtoptop

dxdbottom

bottomtop

dxd

Example

125

2xx

dxd

Another example

5

13

xxy

One more example

xxxxf

7)23(3)(

2

More trig rules

xdxd tan x

dxd sec

Still more

xdxd cot x

dxd csc

xdxd tan x

dxd sec

Try…

xxy

sincos1

xxxf tansec)(

Higher Order DerivativesYou can take more than one

derivative

Position function Velocity function Acceleration function

)()4('''''' nyyyyy

Example

xxy cos2

Find y ‘’