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

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:

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

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 a^x:

Explanation

This proof of the derivative of a^x 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)