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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.

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Exponential Functions

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  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:

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