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 ‘’
