**Derivatives**

**Derivatives**

Definition of a Derivative:

The derivative of the function

f

with the respect to the variable

x

is the function

f'

whose value at

x

is

provided that the limit exists.

Power Rule

Product Rule

Quotient Rule

lo de hi - hi de lo

--------------------

lo squared

Chain Rule

1. Corner

A function will not have a derivative at a point

P(a,f(a))

where the slopes of the secant lines,

fail to approach

a

limit as

x

approaches

a

.

2.Cusp

3.Vertical Tangent

4. Discontinuity

-Jump (piecewise)

-Removable (hole)

-Infinite (asymptote)

where the one-sided derivatives differ

where the slopes of the secant lines approach infinity from one side and negative infinity from the other side

where the slopes of the secant lines approach either infinity or negative infinity from both sides

cause one or both of the one-sided derivatives to be nonexistent

take the exponent and multiply it with the coefficient

take the exponent and subtract one

derivative = 1st term(

u

) * derivative of 2nd term(

dv

) + 2nd term (

v

)* derivative of 1st term(

du

)

1st term

derivative of 1st term

2nd term

derivative of 2nd term

denominator squared

derivative of the denominator

numerator

derivative of nurmerator

(denominator (

v

) * derivative of numerator(

du

) - (numerator(

u

) * derivative of numerator(

dv

)

------------------------------------------------------------------------------------------------------------------

denominator squared

little trick:

derivative= power rule * derivative of what is inside the ()

power rule

derivative of ()

Example of Infinte:

denominator