Basic DerivativesBasic Derivatives

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Brought To You By:Brought To You By:

►Algebraically, theAlgebraically, the►Geometrically as the slope of the Geometrically as the slope of the

tangent line to the graph of at tangent line to the graph of at ►Functionally as the instantaneous Functionally as the instantaneous

rate of change of at rate of change of at

Definition Of A DerivativeDefinition Of A DerivativeThe derivative of a function at a point The derivative of a function at a point

x=ax=a, can be interpreted in several , can be interpreted in several different ways:different ways:

)(xf ax

ax

h

xfhxfh

)()(lim

0

Applications of DerivativesApplications of Derivatives

► Finding the instantaneous velocity of an object at a Finding the instantaneous velocity of an object at a precise moment in timeprecise moment in time

► Finding the instantaneous rate of change of a functionFinding the instantaneous rate of change of a function► Finding the slope of the tangent to the graph of a Finding the slope of the tangent to the graph of a

functionfunction

* NOTE: the derivative symbol can be written as or or* NOTE: the derivative symbol can be written as or ordx

dy)(' xf 'y

Basic Derivative FormulasBasic Derivative Formulas

► If , where n is a constant, then If , where n is a constant, then

► If , where c is a constant, then If , where c is a constant, then

► If , where c is a constant, then If , where c is a constant, then

Power Rule

Constant Rule

Power Rule

If , thenIf , then

If , then If , then

More basic derivative More basic derivative formulasformulas

ueuf )(

uuf ln)( u

uu

du

d '][ln

uu euedu

d'][

Logarithm Rule

Exponential Rule

ExamplesExamples

Solution: Solution: Following the Power Rule, Following the Power Rule,

we can now calculate the derivative.we can now calculate the derivative.

4)( xxf Differentiate:Differentiate:

144)(' xxf34)(' xxf

4)( xxf

Examples (cont.)Examples (cont.)

Solution: Solution: Following the Constant Rule,Following the Constant Rule, we can now we can now

calculate the derivative.calculate the derivative.

Differentiate:Differentiate:

10)( xf

0)(' xf

Examples (cont.)Examples (cont.)

Solution: Solution: Following the Power Rule, we can now calculate the Following the Power Rule, we can now calculate the

derivative.derivative.

Differentiate:Differentiate:

xxf 5)(

xxf 5)(

11)1(5)(' xxf 05)(' xxf

5)(' xf

Examples (cont.)Examples (cont.)

Solution: Solution: Following the Logarithm Rule, we can now Following the Logarithm Rule, we can now

calculate the derivative.calculate the derivative.

Differentiate:Differentiate:

xxf ln)(

xxf ln)(

xxf

1)('

u = xu’ = 1

Examples (cont.)Examples (cont.)

Solution: Solution:

Following the Exponential Rule,Following the Exponential Rule, we can now we can now calculate the derivative.calculate the derivative.

Differentiate:Differentiate:xexf 2)(

xexf 2)( xu 22'u

xexf 22)('

Questions???Questions???

Helpful LinksHelpful Links

►Derivatives and Integrals HandoutDerivatives and Integrals Handout► Implicit Differentiation HandoutImplicit Differentiation Handout►Derivatives Student HandoutDerivatives Student Handout►Derivatives QuizDerivatives Quiz