Post on 14-Dec-2015
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)('