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**Unformatted text preview: **Lecture 27 1 Lecture 27: (Section 5.4) (OMIT 5.3) Differentiation of Exponential Functions Remember that for function y = f ( x ), dy dx = f ( x ) = What if f ( x ) = e x ? First, recall that we defined e = lim m (1 + 1 m ) m Lecture 27 2 We have another important limit: lim h e h 1 h = NOTE: Table, p. 361 We can also see the limit by looking at the graph of y = e x 1 x : Minus 1.0 Minus 0.5 0.0 0.5 1.0 0.5 1.0 1.5 2.0 Lecture 27 3 Now we can find f ( x ) for f ( x ) = e x : We have the following result: If f ( x ) = e x , then f ( x ) = Lecture 27 4 ex. Find the slope of the tangent line to f ( x ) = e x ( x 1) at x = ln2. Lecture 27 5 ex. Find the relative extrema of f ( x ) = e x x 2 + 1 . Lecture 27 6 We can also use the Chain Rule to differentiate: If f is a differentiable function, then d dx ( e f ( x ) ) = ex. Find f ( x ) for the following: 1) f ( x ) = e x 3 3 x +3 Lecture 27 7 2) f ( x ) = e 2 x +1 Then write the equation of the tangent line to y = f ( x ) at x = 0. Lecture 27...

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