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AP Calculus AB: 5.3.2 The Derivative of the Natural Log Function

Mathematics10 CardsCreated 3 months ago

This flashcard set explores how to find derivatives involving natural logarithms using the chain rule and properties of logarithms. It connects the derivative of ln π‘₯ lnx to the exponential function and includes applications with composite and complex logarithmic expressions, reinforcing both conceptual understanding and computational skills.

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The Derivative of the Natural Log Function

  • The derivative of the natural exponential function is itself. Use the chain rule to find the derivative of the composition of the natural exponential function and another function.

  • The natural exponential function can be used to define the derivative of the natural log function. Use the chain rule to find the derivative of the composition of the natural log function and another function.

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Key Terms

Term
Definition

The Derivative of the Natural Log Function

  • The derivative of the natural exponential function is itself. Use the chain rule to find the derivative of the composition of the natural e...

note

  • You can find the derivative of the natural log function if
    you know the derivative of the natural exponential function.

  • C...

Find the x values of the points where the graphs of y=3^2x+1 and y=4^x intersect.

xβ‰ˆβˆ’1.35

Find dy/dx, where y=(x2+2)[ln(x^2+1)]

dy/dx=2x(x^2+2) / x^2+1 +2xln(x^2+1)

Find dy/dx, where y=log2(x^2+1).

dy/dx=2x /ln2β‹…(x^2+1)

Find the derivative of f (x) = lnβ€Š(ln 3x).

fβ€²(x)=1/xln(3x)

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AP Calculus AB: 5.3.2 The Derivative of the Natural Log Function
TermDefinition

The Derivative of the Natural Log Function

  • The derivative of the natural exponential function is itself. Use the chain rule to find the derivative of the composition of the natural exponential function and another function.

  • The natural exponential function can be used to define the derivative of the natural log function. Use the chain rule to find the derivative of the composition of the natural log function and another function.

note

  • You can find the derivative of the natural log function if
    you know the derivative of the natural exponential function.

  • Consider the function given by the number e raised to the power ln x. The log identities prove that this expression is equal to x.

  • Take the derivative of both equations. Notice that the
    derivatives must be equal. Moreover, the derivative of this expression includes the derivative of the natural log function as one of its factors.

  • Isolate that factor. The result is the derivative of the natural logarithmic function.

  • You can use a similar process to find the derivative of any log function.

  • You will often need to use the chain rule when finding the derivative of a log function.

  • Think of the argument of the log as a blop. The chain rule states that you must multiply the result by the derivative of the blop.

  • Notice that you can use the properties of logs to make finding some derivatives easier.

Find the x values of the points where the graphs of y=3^2x+1 and y=4^x intersect.

xβ‰ˆβˆ’1.35

Find dy/dx, where y=(x2+2)[ln(x^2+1)]

dy/dx=2x(x^2+2) / x^2+1 +2xln(x^2+1)

Find dy/dx, where y=log2(x^2+1).

dy/dx=2x /ln2β‹…(x^2+1)

Find the derivative of f (x) = lnβ€Š(ln 3x).

fβ€²(x)=1/xln(3x)

Suppose f (x) = ln x^2. Find f β€˜(x).

fβ€²(x)=2/x

Given y=ln(√x ^3√x+2 ^7√3xβˆ’1),find dy/dx.

dy/dx=1/2x+1/ 3x+6 + 3/21xβˆ’7

Find fβ€²(x) if f(x)=(lnx)^2.

fβ€²(x)=2lnx/x

Suppose you are given the function f(x)=ln(e^4). Find fβ€²(x).

fβ€²(x)=0