Derivatives of Exponential & Logarithmic Fuctions

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Derivs of Exponents & Logs: Exponentials | Logs | Logarithmic Diff. | Practice Problems | Quiz

On this page, we'll discover how differentiate exponential and logarithmic functions. We'll also look at the technique of logarithmic differentiation, which can be used to find some difficult derivatives.

DERIVATIVES OF EXPONENTIAL FUNCTIONS: (top)

We'll divide our study of the derivatives of exponential functions into two parts: where the base is e, and where the base is something else.

1. Derivative of ex:

Explanation
We'll use the definition of derivative:
=
by the definition of derivative
 
=
by the multiplication rule for exponents
 
=
by factoring out ex
 
=
by the constant rule for limits (since ex is constant with respect to h)
We now need to evaluate . To do this, we'll go back for a moment to the definition of e:
e
= by definition
  = Note the equivalence of these two forms by observing that in both cases the exponent aproaches and the x-term within the parentheses approaches 0.
Thus, by substitution
Substitutng this back in, we get:
= by subistitution
  = by the power rule of exponents
  = simplification
  = simplification
  = ex by evaluating the limit with the constant rule

Note that this means by the chain rule, .
PROBLEM 1:
Find the derivative of .
Explanation
=
by the derivative of ex and the chain rule
 
=
by the derivative of sin(x) and the chain rule
 
=
by the power rule

2. Derivative of ax:

Explanation
This proof of the derivative of ax requries knowing that . Review the derivative of ln(x) here.

DERIVATIVES OF LOGARITHMIC FUNCTIONS: (top)

1. Derivative of Natural Logs:

Explanation

2. Derivative of Other Logs:

Explanation
=
by the change of base formula for logarithms
=
by rewriting the division as multiplication
=
by the constant multiple rule, since is a constant
=
by the derivative of ln(x)

LOGARITHMIC DIFFERENTIATION: (top)

 

PRACTICE PROBLEMS: (top)

QUIZ: (top)

 

This page last updated 10 August, 2008 6:23 PM