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

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 ‘’