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

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• Pg. 356 1, 3, 9, 11, 35-51 odd, 57, 77, 79, 85-93 odd, 99, 101