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5.4 Exponential Functions: Differentiation and Integration

5.4 Exponential Functions: Differentiation and Integration. The inverse of f(x) = ln x is f -1 = e x . Therefore, ln (e x ) = x and e ln x = x. Solve for x in the following equations. Take the ln of both sides. Operations with Exponential Functions.

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5.4 Exponential Functions: Differentiation and Integration

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  1. 5.4 Exponential 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.

  2. Operations with Exponential Functions The Derivative of the Natural Exponential Function Differentiate.

  3. Find the relative extrema of Since ex never = 0, -1 is the only critical number. neg. pos. Therefore, x = -1 is a min. by the first derivative test. -1 dec. inc. Minimum @ ?

  4. Integration Rules for Exponential Functions Ex. Let u = 3x + 1 du = 3 dx

  5. Ex. Let u = -x2 du = -2x dx Ex. Let u = 1/x = x-1

  6. Let u = cos x Ex. du = -sin x dx Let u = -x du = -dx -du = dx Ex.

  7. Ex. Ex. Let u = ex du = ex dx

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