derivative-of-exponential-and-logarithmic-functions.pdf PDF

Worksheet: Logarithmic Function

Write the following equalities in exponential form. (1) log3 81 = 4. (2) log7 7 = 1. (3) log1. 2.

Logarithmic Equations.pdf

Worksheet by Kuta Software LLC. Kuta Software - Infinite Algebra 2. Name___________________________________. Period____. Date________________. Logarithmic

Worksheet 2.7 Logarithms and Exponentials

Logs have some very useful properties which follow from their definition and the equivalence of the logarithmic form and exponential form. Some useful

Infinite Precalculus - Exponents and Logarithms

. Worksheet by Kuta Software LLC. Kuta Software - Infinite Precalculus. Exponents and Logarithms Rewrite each equation in logarithmic form. 13) x y = 178. 14 ...

Integrals Involving Exponential and Logarithmic Functions

In this section we explore integration involving exponential and logarithmic functions. Integrals of Exponential Functions. The exponential function is perhaps

Exponential and logarithmic functions

The module Indices and logarithms (Years 9–10) covered many properties of exponential and logarithmic functions including the index and logarithm laws. Now

Derivatives of Exponential and Logarithmic Functions. Logarithmic

Derivatives of Exponential and Logarithmic Functions. Logarithmic Differentiation. Derivative of exponential functions. The natural exponential function can

Math 120 - Review Sheet Exponential and Logorithmic Functions

Worksheet by Kuta Software LLC. Rewrite each equation in logarithmic form. 16) 4. 2. = 16. 17) x. −4. = y. 18) m. 3. = n. 19) 12 x. = y. 20) a. −7. = b. Find

Infinite Algebra 2 - Solving Exponential and Logarithmic Equations

. Worksheet by Kuta Software LLC. -2-. Solve each equation. Round your answers to the nearest ten-thousandth. 11) 4 x. = 72. 12) e b - 2. = 12. 13) e r. - 7 =

Unit 8: Exponential & Logarithmic Functions

Worksheet. 13. Review. 14. Test. Date ______. Period_________. Unit 8: Exponential & Logarithmic Functions. Page 2. Objective: To model exponential growth.

Worksheet: Logarithmic Function

Write the following equalities in exponential form. (1) log3 81 = 4. (2) log7 7 = 1. (3) log1. 2.

derivative-of-exponential-and-logarithmic-functions.pdf

If you are not familiar with exponential and logarithmic functions you may wish to consult the booklet Exponents and Logarithms which is available from the

Worksheet 2.7 Logarithms and Exponentials

Logs have some very useful properties which follow from their definition and the equivalence of the logarithmic form and exponential form. Some useful

Higher Mathematics EXPONENTIALS & LOGARITHMS

4 Exponentials and Logarithms to the Base e. EF 6. 5 Exponential and Logarithmic Equations Remember that the graph of an exponential function ( ).

Indiana Landmarks

Write an example below to some of exponents worksheet functions showing Exponential functions date either Work logarithmic function. Algebra worksheets ...

Unit 8: Exponential & Logarithmic Functions

8.5 Exponential and Logarithmic. Equations Applications of Natural Logs. Worksheet ... B. An exponential function is a function with the general form ...

Exponential and Logarithmic Graph Worksheet

Graphing Exponential and Logarithmic Functions. Name: Graph the following Exponential Functions. 1. 3. 2. 1 x y. +. = -. 2. 2. 4. 3. 1. 3 x y. -. ? ?. = +.

Comparing Exponential and Logarithmic Rules

Task 1: Looking closely at exponential and logarithmic patterns… 1) In a prior lesson you graphed and then compared an exponential function with a

6.5 Applications of Exponential and Logarithmic Functions

Equations 6.2 and 6.3 both use exponential functions to describe the growth of an investment. Curiously enough the same principles which govern compound

Limits of exponential and logarithmic functions worksheet

Limits of exponential logarithmic and trigonometric functions worksheet. Applying the natural logarithm function to both sides of the equation ...

Mathematics Learning Centre

Derivatives of exponential and

logarithmic functions

Christopher Thomas

c?1997 University of Sydney Mathematics Learning Centre, University of Sydney1

1Derivatives of exponential and logarithmic func-

tions If you are not familiar with exponential and logarithmic functions you may wish to consult the bookletExponents and Logarithmswhich is available from the Mathematics Learning

Centre.

Youmayhave seen that there are two notations popularly used for natural logarithms, log e and ln. These are just two different ways of writing exactly the same thing, so that log e x≡lnx.Inthis booklet we will use both these notations.

The basic results are:

d dxe x =e x d dx(log e x)=1 x. Wecan use these results and the rules that we have learnt already to differentiate functions which involve exponentials or logarithms.

Example

Differentiate log

e (x 2 +3x+1).

Solution

Wesolve this by using the chain rule and our knowledge of the derivative of log e x. d dxlog e (x 2 +3x+1) =d dx(log e u)(whereu=x 2 +3x+1) d du(log e u)×du dx(by the chain rule) 1 u×dudx 1 x 2 +3x+1×ddx(x 2 +3x+1) 1 x 2 +3x+1×(2x+3) 2x+3 x 2 +3x+1.

Example

Find d dx (e 3x 2 Mathematics Learning Centre, University of Sydney2

Solution

This is an application of the chain rule together with our knowledge of the derivative of e x d dx(e 3x 2 )=de u dxwhereu=3x 2 =de u du×dudxbythe chain rule =e u

×du

dx =e 3x 2 ×d dx(3x 2 =6xe 3x 2

Example

Find d dx (e x 3 +2x

Solution

Again, we use our knowledge of the derivative ofe

x together with the chain rule. d dx(e x 3 +2x )=de u dx(whereu=x 3 +2x) =e u

×du

dx(by the chain rule) =e x 3 +2x ×d dx(x 3 +2x) =(3x 2 +2)×e x 3 +2x

Example

Differentiate ln(2x

3 +5x 2 -3).

Solution

Wesolve this by using the chain rule and our knowledge of the derivative of lnx. d dxln(2x 3 +5x 2 -3) =dlnu dx(whereu=(2x 3 +5x 2 -3) dlnu du×dudx(by the chain rule) 1 u×dudx 1 2x 3 +5x 2 -3×ddx(2x 3 +5x 2 -3) 1 2x 3 +5x 2 -3×(6x 2 +10x) 6x 2 +10x 2x 3 +5x 2 -3. Mathematics Learning Centre, University of Sydney3 There are two shortcuts to differentiating functions involving exponents and logarithms.

The four examples above gave

d dx(log e (x 2 +3x+1)) =2x+3 x 2 +3x+1 d dx(e 3x 2 )=6xe 3x 2 d dx(e x 3 +2x )=(3x 2 +2)e 3x 2 d dx(log e (2x 3 +5x 2 -3)) =6x 2 +10x 2x 3 +5x 2 -3.

These examples suggest the general rules

d dx(e f(x) )=f (x)e f(x) d dx(lnf(x)) =f (x) f(x). These rules arise from the chain rule and the fact that de x dx =e x and dlnx dx 1 x .They can speed up the process of differentiation but it is not necessary that you remember them. If you forget, just use the chain rule as in the examples above.

Exercise 1

Differentiate the following functions.

a.f(x)=ln(2x 3 )b.f(x)=e x 7 c.f(x)=ln(11x 7 d.f(x)=e x 2 +x 3 e.f(x)=log e (7x -2 )f.f(x)=e -x g.f(x)=ln(e x +x 3 )h.f(x)=ln(e x x 3 )i.f(x)=ln x 2 +1 x 3 -x Mathematics Learning Centre, University of Sydney4

Solutions to Exercise 1

a.f (x)=6x 2 2x 3 =3 x

Alternatively writef(x)=ln2+3lnxso thatf

(x)=31 x. b.f (x)=7x 6 e x 7 c.f (x)= 7 x d.f (x)=(2x+3x 2 )e x 2 +x 3 e.Writef(x)=log e

7-2log

e xso thatf (x)=- 2 x f.f (x)=-e -x g.f (x)=e x +3x 2 e x +x 3 h.Writef(x)=lne x 3 lnx so thatf (x)=1+3 x. i.Writef(x)=ln(x 2 +1)-ln(x 3 -x)sothatf (x)=2x x 2 +1-3x 2 -1 x 3 -x.quotesdbs_dbs12.pdfusesText_18

[PDF] derivative-of-exponential-and-logarithmic-functions.pdf

Mathematics Learning Centre

Derivatives of exponential and

Christopher Thomas

1Derivatives of exponential and logarithmic func-

Centre.

The basic results are:

Example

Differentiate log

Solution

Example

Solution

×du

Example

Solution

Again, we use our knowledge of the derivative ofe

×du

Example

Differentiate ln(2x

Solution

The four examples above gave

These examples suggest the general rules

Exercise 1

Differentiate the following functions.

Solutions to Exercise 1

Alternatively writef(x)=ln2+3lnxso thatf

7-2log