SECTION 5-5A Part I: Exponentials base other than e.

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Presentation transcript:

SECTION 5-5A Part I: Exponentials base other than e

Variables in the exponent Logarithmic differentiation is used to find derivatives of exponential functions: 1) Which if these is an exponential function?

Definition of exponential function with base a Let a be a positive real number where a ≠1 and let x be any real number, then the exponential function to the base a is defined by: Bases other than e

Properties a) b) c) d)

Definition of logarithmic function with base a Let a be a positive real number where a ≠1 and let x be any real number, then the logarithmic function to the base a is defined by: Change of base formula

Properties a) b) c) d)

Differentiate

Derivatives for bases other than e Let a be a positive real number where a ≠1 and let u be a differentiable function of x such that: I. II.

Derivatives for bases other than e Let a be a positive real number where a ≠1 and let u be a differentiable function of x such that: III. IV. Proof pg 364

Differentiate

PRODUCT RULE Differentiate

Careful: not the power rule!

Differentiate

Through the magic of math:

Assignment Page 368 # 41-48, , , and 67 all