5.2.2 Derivatives of Exponential Functions

Question 1

Derivatives of Exponential Functions

Answer 1

• Studying the slopes of tangent lines to the graph of a function can help you determine the derivative.
• The derivative of an exponential function is the product of the function and the natural log of its base.

Question 2

note

Answer 2

• Before trying to find the derivative of an
  exponential function, it is a good idea to examine
  the behavior of the lines tangent to the function.
• Since the base is greater than 1, the tangent lines
  are always positive. In addition, the slopes of the
  tangent lines seem to be increasing.
• Use the definition of the derivative to find the
  derivative of a general exponential function.
• Notice that you can factor the exponential function
  out of the limit. You can do this because there are
  no ∆ x-terms in that factor.
• The remaining factor that includes the limit portion is
  harder to evaluate. However, it is apparent that the
  derivative of the exponential function is equal to that
  same exponential function times the result of that
  limit.
• Even though you cannot solve the limit directly, you
  can approximate the value of the slope for an
  exponential function at a point.
• Remember, smaller values of ∆ x will give you better
  approximations of the value of the limit.
• It turns out that the limit in question is equal to the
  natural log of the base of the exponential function.
• Therefore the derivative of the natural exponential
  function is itself, since the natural log of e is 1.
• Here is the formula for the derivative of the general
  exponential function.

Question 3

For what values of x does the function h(x)=4e^4x−16x have negative derivatives?

Answer 3

x < 0

Question 4

For what values of x does the graph of g(x)=e^2x+1 have horizontal tangent lines?

Answer 4

There are no values of x which correspond to horizontal tangent lines.

Question 5

Consider the function R(x)=N^x, where N>1. Suppose you are told that R′(x)=N^x.For what values of N is this possible?

Answer 5

N = e

Question 6

Find the derivative of the function f(x)=2^x.

Answer 6

f′(x)=2^x⋅ln2

Question 7

Compute the derivative of the function f(x)=2x^2+2e^x

Answer 7

f′(x)=4x+2e^x

Question 8

Let N be a positive number with N > 1. What can you conclude about the derivative of the function f (x) = N x using its graph?

Answer 8

The derivative of f (x) increases as x increases.