## 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)