- Explanation
- We'll use the definition of derivative:

=by the definition of derivative =by the multiplication rule for exponents =by factoring out e ^{x}=by the constant rule for limits (since e ^{x}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 = **e**^{x}by evaluating the limit with the constant rule - Explanation
- =
by the derivative of and the chain rule**e**^{x}=by the derivative of sin( *x*) and the chain rule=by the power rule - Explanation
- This proof of the derivative of a
requries knowing that . Review the derivative of ln(^{x}*x*) here.

- Explanation
- 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*)

## Derivatives of Exponential & Logarithmic Fuctions

Home > Calculus 1 > Derivs of Exponents & Logs

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 e^{x}:

Note that this means by the chain rule, .

PROBLEM 1:

Find the derivative of .

2. Derivative of a^{x}:

**DERIVATIVES OF LOGARITHMIC FUNCTIONS:** (top)

1. Derivative of Natural Logs:

2. Derivative of Other Logs:

**LOGARITHMIC DIFFERENTIATION:** (top)

**PRACTICE PROBLEMS:** (top)

**QUIZ:** (top)

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