- 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
by the derivative of ex and the chain rule= by the derivative of sin(x) and the chain rule= by the power rule
- This proof of the derivative of ax requries knowing that . Review the derivative of ln(x) here.
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
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:
Note that this means by the chain rule, .
Find the derivative of .
2. Derivative of ax:
1. Derivative of Natural Logs:
2. Derivative of Other Logs:
This page last updated 10 August, 2008 6:23 PM