1 / 12

Exponential Functions

Exponential Functions. If a > 0, a ≠ 1, f ( x ) = a x is a continuous function with domain R and range (0, ∞ ). In particular, a x > 0 for all x . If 0 < a < 1, f ( x ) = a x is a decreasing function . If a > 1, f is an increasing function.

gareth-boone
Download Presentation

Exponential Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Exponential Functions • If a > 0, a≠ 1, f(x) = ax is a continuous function with domain Rand range (0, ∞). • In particular, ax > 0 for all x. • If 0 < a < 1, f(x) = ax is a decreasing function. • Ifa > 1, f is an increasing function. There are basically three kinds of exponential functions y = ax. Since (1/a)x=1/ax= a-x, the graph of y =(1/a)x is just the reflection of the graph of y = axabout the y-axis. The exponential function occurs very frequently in mathematical models of nature and society.

  2. Derivative of exponential function By definition, this is derivative , what is the slope of at .

  3. example: y = etan x Differentiate the function • To use the Chain Rule, we let u = tan x. • Then, we have y = eu. example: Find y’ if y = e-4x sin 5x.

  4. chain rule: We can now use this formula to find the derivative of

  5. Derivative of Natural Logarithm Function

  6. example: Differentiate y = ln(x3 + 1). • To use the Chain Rule, we let u = x3+1. • Then, y =ln u. example: Find:

  7. Differentiate example: example: If we first simplify the given function using the laws of logarithms, the differentiation becomes easier

  8. example: Find f ’(x) if • Thus, f ’(x) = 1/x for all x ≠ 0. The result is worth remembering:

  9. Derivative of Logarithm Function a logarithmic function with base a in terms of the natural logarithmic function: Since lna is a constant, we can differentiate as follows: example:

  10. IMPORTANT and UNUSUAL: If you have a daunting task to find derivative in the case of a function raised to the function, or a crazy product, quotient, chain problem you do a simple trick: FIRST find logarithm so you’ll have sum instead of product, and product instead of exponent. Life will be much, much easier. STEPS IN LOGARITHMIC DIFFERENTIATION • Take natural logarithms of both sides of an equation y = f(x) and use the Laws of Logarithms to simplify. • Differentiate implicitly with respect to x. • Solve the resulting equation for y’.

  11. example: Differentiate: 1. 2. 3. Since we have an explicit expression for y, we can substitute and write If we hadn’t used logarithmic differentiation the resulting calculation would have been horrendous.

  12. example: Try:

More Related