5.4 exponential functions: differentiation and integration
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5.4Exponential Functions:
Differentiation and Integration
The inverse of f(x) = ln x is f-1 = ex.
Therefore, ln (ex) = x and e ln x = x
Solve for x in the following equations.
Take the ln of both sides.
Operations with Exponential Functions
The Derivative of the Natural Exponential Function
Differentiate.
Find the relative extrema of
Since ex never = 0, -1 is the onlycritical number.
-1
neg.
dec.
pos.
inc.
Therefore, x = -1 is a min. bythe first derivative test.
Minimum @ ?
Integration Rules for Exponential Functions
Ex.Let u = 3x + 1
du = 3 dx
Ex. Let u = -x2
du = -2x dx
Ex. Let u = 1/x = x-1
Ex. Let u = cos x
du = -sin x dx
Ex. Let u = -x du = -dx -du = dx
Ex.
Ex. Let u = ex
du = ex dx
Don’t Want You to Be Bored…
• Pg. 356 1, 3, 9, 11, 35-51 odd, 57, 77, 79, 85-93 odd, 99, 101
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