Derivatives of Exponential Functions

Question 1

what is the derivative of e^x

Answer 1

e^x

Question 2

how do you find the derivative of e^f(x)

Answer 2

the original function multiplied by the derivative of the exponent

Question 3

how to find the coordinates of an exponential function where the tangent line is horizontal?

Answer 3

1. find the derivative
2. find critical values by setting y’=0 and factoring
3. find the y coordinate for each x value by plugging the x value into the original equation

Question 4

how do you find the derivative of an exponential function like y=e^xx^-1?

Answer 4

1. use the product rule; first multiplied by the derivative of the second added to the derivative of the first multiplied by the second
2. factor out the greatest common factor
3. when to variables are added or subtracted, make sure they have the same denominator

Question 5

how do you find the derivative of an exponential function like y=e^x/1-e^2x?

Answer 5

1. use the quotient rule; ho(dehi)-hi(deho)/ho^2
2. factor out the greatest common factor
3. collect like terms in the bracket

Question 6

how do you find the x-intercepts of an exponential function?

Answer 6

set the original equation equal to zero

Question 7

how do you find the y-intercepts of an exponential function?

Answer 7

set x in the original function equal to zero

Question 8

how to find the horizontal asymptote of an exponential function?

Answer 8

use the c value in the exponential function

or

graph the function to see where to horizontal asymtote is

Question 9

how do you find the vertical asymptotes of an exponential functions?

Question 10

how do you find the maximum and minimum values of an exponential function?

Answer 10

1. find the first derivative
2. find the second derivative
3. find critical values by setting the first derivative equal to zero and factoring
4. find the slope of the x value found from the first derivative using the second derivative

• if the answer is negative this wouls be a concave down so label it a maximum

if the answer is positive, this would be a concave up like a cup so label it a maximum point

5. use the original function to find the y value
6. write a therefore statement; there is the maximum and minimum is at ( , )

Question 11

how do you find the points of inflection of an exponential function?

Answer 11

1. find the second derivative of the original function
2. set the second derivative equal to zero and solve for x values
3. plot the x values on a number line labeled y’’.
4. In each segment of the number line, write the number in between that you will use to plug into the second derivative

5.
- if the answer is positive, label the segment with a positive sign

-if the answer is negative, label the segment with a negative sign

-do this for each number in each segment

6. if there are opposite signs in each segment then there is a point of inflection
7. find the y value of each x value on the number line
8. write a, therefore, statement, therefore poi at ( , ) and ( , ).