5.4 Exponential Functions: Differentiation and Integration ...
5.4 Exponential Functions: Differentiation and Integration
Definition of the Natural Exponential Function ? The inverse function of the natural logarithmic function
f x ln x is called the natural exponential function and is denoted by f 1 x ex . That is, y ex if
and only if x ln y .
Properties of the Natural Exponential Function:
1. The domain of f x ex is , , and the range is 0, . 2. The function f x ex is continuous, increasing, and one-to-one on its entire domain. 3. The graph of f x ex is concave upward on its entire domain.
4. limx ex 0 and limx ex
Operations with Exponential Functions ? Let a and b be any real numbers.
1. eaeb eab
2.
ea eb
eab
Examples: Solve for x accurate to three decimal places.
1. eln2x 12
2. 6 3ex 8
3. ln 4x 1
Derivatives of Natural Exponential Functions ? Let u be a differentiable function of x.
1.
d dx
ex
ex
2.
d dx
eu
eu
du dx
u 'eu
Examples: Find the derivative.
1. y e5x
3. y xex
2. y ex2 4. y xxex
5. y ex ex 2
6. y ln ex
Examples: Find the equation of the tangent line to the graph of the function at the given point.
1. y e2xx2 , 2,1
2. f x e3 ln x, 1,0
Example: Use implicit differentiation to find dy/dx given exy x2 y2 10 .
Example: Find the second derivative of g x x ex ln x
Integration Rules for Exponential Functions ? Let u be a differentiable function of x.
1. exdx ex C
2. eudu eu C
Examples: Find the indefinite integral.
1. ex4 4x3 dx
2.
ex
ex
1
2
dx
3.
e2x
2ex ex
1dx
Examples: Evaluate the definite integral.
1. 4 e3xdx 3
1 ex
2. 0 5 ex dx
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