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Derivatives of Logarithmic and Exponential Functions- DS2

Objectives

  • Find the derivative of logarithmic functions.
  • Use logarithmic differentiation to determine the derivative of a function.

Summary

We have stated a rule for derivatives of exponential functions in the same spirit as the rule for power functions: for any positive real number a , if f ( x ) = a x , then f ′ ( x ) = a x ln ⁡ ( a ) .

For an exponential function f ( x ) = a x ( a > 1 ) , the graph of f ′ ( x ) appears to be a scaled version of the original function. In particular, careful analysis of the graph of f ( x ) = 2 x , suggests that d d x [ 2 x ] = 2 x ln ⁡ ( 2 ) , which is a special case of the rule we stated in Section 2.1.

In what follows, we find a formula for the derivative of g ( x ) = ln ⁡ ( x ) . To do so, we take advantage of the fact that we know the derivative of the natural exponential function, the inverse of g . In particular, we know that writing g ( x ) = ln ⁡ ( x ) is equivalent to writing e g ( x ) = x . Now we differentiate both sides of this equation and observe that

d d x [ e g ( x ) ] = d d x [ x ] .

The right hand side is simply 1 ; by applying the chain rule to the left side, we find that

e g ( x ) g ′ ( x ) = 1 .

Next we solve for g ′ ( x ) , to get

g ′ ( x ) = 1 e g ( x ) .

Finally, we recall that g ( x ) = ln ⁡ ( x ) , so e g ( x ) = e ln ⁡ ( x ) = x , and thus

g ′ ( x ) = 1 x .

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