section 3.1 derivatives of polynomials and exponential functions goals learn formulas for the...

Section 3.1Section 3.1

Derivatives of Polynomials and Derivatives of Polynomials and Exponential FunctionsExponential Functions

GoalsGoals• Learn formulas for the derivatives ofLearn formulas for the derivatives of

ConstantConstant functions functions PowerPower functions functions ExponentialExponential functions functions

• Learn to find new derivatives from old:Learn to find new derivatives from old: Constant Constant multiplesmultiples SumsSums and and differencesdifferences

Constant FunctionsConstant Functions

The graph of the constant function The graph of the constant function ff((xx) = ) = cc is the horizontal line is the horizontal line yy = = c c ……• which has slope 0 ,which has slope 0 ,• so we must have so we must have f f ((xx) = 0 (see the next ) = 0 (see the next

slide).slide). A formal proof is easy:A formal proof is easy:

Constant Functions (cont’d)Constant Functions (cont’d)

Power FunctionsPower Functions

Next we look at the functions Next we look at the functions ff((xx) = ) = xxnn , where , where nn is a positive integer. is a positive integer.

If If nn = 1 , then the graph of = 1 , then the graph of ff((xx) = ) = xx is the line is the line yy = = xx , which has slope , which has slope 1 , so 1 , so f f ((xx) = 1.) = 1.

We have already seen the cases We have already seen the cases nn = = 2 and 2 and nn = 3 : = 3 :

Power Functions (cont’d)Power Functions (cont’d) For For nn = 4 we find the derivative of = 4 we find the derivative of

ff((xx) = ) = xx44 as follows: as follows:

Power Functions (cont’d)Power Functions (cont’d)

There seems to be a pattern There seems to be a pattern emerging!emerging!

It appears that in general, if It appears that in general, if ff((xx) = ) = xxnn , then, then

f f ((xx) = ) = nxnxn n -- 11 . . This turns out to be the case:This turns out to be the case:


We illustrate the Power Rule using a We illustrate the Power Rule using a variety of notations:variety of notations:

It turns out that the Power Rule is It turns out that the Power Rule is valid for valid for anyany real number real number nn , not just , not just positive integers:positive integers:

Constant MultiplesConstant Multiples The following formula says that The following formula says that the the

derivative of a constant times a function derivative of a constant times a function is the constant times the derivative of is the constant times the derivative of the functionthe function::

Sums and DifferencesSums and Differences

These next rules say that These next rules say that the the derivative of a sum (difference) of derivative of a sum (difference) of functions is the sum (difference) of functions is the sum (difference) of the derivativesthe derivatives::

ExampleExample

Exponential FunctionsExponential Functions

If we try to use the definition of derivative If we try to use the definition of derivative to find the derivative of to find the derivative of ff((xx) = ) = aaxx , we get: , we get:

The factor The factor aaxx doesn’t depend on doesn’t depend on xx , so , so we can take it in front of the limit:we can take it in front of the limit:

Exponential (cont’d)Exponential (cont’d)

Notice that the limit is the value of Notice that the limit is the value of the derivative of the derivative of ff at 0 , that is, at 0 , that is,


This shows that…This shows that…• ifif the exponential function the exponential function ff((xx) = ) = aaxx is is

differentiable at 0 ,differentiable at 0 ,• thenthen it is differentiable everywhere and it is differentiable everywhere and

f f ((xx) = ) = f f (0)(0)aaxx

Thus, Thus, the rate of change of any the rate of change of any exponential function is proportional exponential function is proportional to the function itselfto the function itself..


The table shown gives numerical The table shown gives numerical evidence for the existence of evidence for the existence of f f (0) (0) whenwhen• aa = 2 ; here apparently = 2 ; here apparently

f f (0) ≈ 0.69(0) ≈ 0.69• aa = 3 ; here apparently = 3 ; here apparently

f f (0) ≈ 1.10(0) ≈ 1.10


So there should be a number So there should be a number aa between 2 and 3 for which between 2 and 3 for which f f (0) = 1 (0) = 1 , that is,, that is,

But the number But the number ee introduced in introduced in Section 1.5 was chosen to have just Section 1.5 was chosen to have just this property!this property!

This leads to the following definition:This leads to the following definition:

0

1lim 1

h

h

ah


Geometrically, this means thatGeometrically, this means that• of all the exponential functions of all the exponential functions yy = = aaxx , ,• the function the function ff((xx) = ) = eexx is the one whose is the one whose

tangent at (0, 1) has a slope tangent at (0, 1) has a slope f f (0) that (0) that is exactly 1 .is exactly 1 .

• This is shown on the next slide:This is shown on the next slide:


This leads to the following This leads to the following differentiation formula:differentiation formula:

Thus, Thus, the exponential function the exponential function ff((xx) = ) = eexx is its own derivative is its own derivative..

ExampleExample

If If ff((xx) = ) = eexx – – xx , find , find ff ((xx) and ) and f f (0) .(0) .

SolutionSolution The Difference Rule gives The Difference Rule gives

ThereforeTherefore

Solution (cont’d)Solution (cont’d)

Note that Note that eexx is positive for all is positive for all xx , so , sof f ((xx) > 0 for all ) > 0 for all xx . .

Thus, the graph ofThus, the graph offf is concave up. is concave up.• This is confirmedThis is confirmed

by the graph shown.by the graph shown.

ReviewReview

Derivative formulas for polynomial Derivative formulas for polynomial and exponential functionsand exponential functions

Sum and Difference RulesSum and Difference Rules The natural exponential function The natural exponential function eexx

section 3.1 derivatives of polynomials and exponential functions goals learn formulas for the...

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