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AP Calculus AB: 5.2.2 Derivatives of Exponential Functions

Mathematics8 CardsCreated 8 months ago

This flashcard set focuses on the rules for differentiating exponential functions, emphasizing how the derivative of 𝑎𝑥 ax involves multiplying by the natural logarithm of the base. It includes both conceptual insights (like tangent line behavior) and practical derivative calculations, reinforcing key calculus principles through graphical and algebraic examples.

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Derivatives of Exponential Functions

  • 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.

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Key Terms

Term
Definition

Derivatives of Exponential Functions

  • Studying the slopes of tangent lines to the graph of a function can help you determine the derivative.

  • The derivative of an ...

note

  • Before trying to find the derivative of an
    exponential function, it is a good idea to examine
    the behavior of the lines tangent to th...

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

x < 0

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

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

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?

N = e

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

f′(x)=2^x⋅ln2

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AP Calculus AB: 5.2.2 Derivatives of Exponential Functions
TermDefinition

Derivatives of Exponential Functions

  • 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.

note

  • 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.

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

x < 0

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

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

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?

N = e

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

f′(x)=2^x⋅ln2

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

f′(x)=4x+2e^x

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?

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