## Lecture Notes: Wednesday, October 28, 1998

Eighth Homework
Assignment - for discussion at recitations on Tuesday November 3, and
Thursday November 5.

### Chapter 5

**Section 4 (Differentiation of Exponential
Functions)**
**Section 5 (Differentiation of Logarithmic
Functions)**

FACT: as h -> 0, (e^h - 1)/h -> 1.

From this we derived the fact that the **derivative of the
exponential function** (base e) is the exponential function
itself.

(d/dx)(exp(x) = exp(x).

Differentiate the functions: e^(x^2), e^(2*x), 2^x, e^ln(x)

Now we can do max/min, graphing, tangent (etc.) problems with
exponential and logarithmic functions. Be sure to work the assigned
problems from Old Final Exams.

**Inverse Functions** again. f(g(x)) = x, g(f(x)) = x.
Differentiating both sides of the latter gives: g'(f(x)) = 1/f'(x).
We use this technique to get a formula for the derivative of
ln(x).

Differentiating x = exp(ln(x)) gives: 1 = exp(ln(x))*ln'(x), from
which we determined

(d/dx)(ln(x) = 1/x.

Examples: Differentiate: ln(x^2+2), log[10](x) (In doing
the second one we derived the fact that

log[10](e) = 1/ln(10).

**Logarithmic differentiation**. Use it on

- (x^2+1)(x-4)(x^3+3x)(x)
- x^(x^2)

In MAPLE, we looked at the graphs of e^x, e^(-x), and
e^(-x^2).

***********************

EXAM #3 Wednesday, October 28. We are in DRL A1 and A8. Please be in
your seats by 6:20.

***********************

Maple Demonstration
#8 Please work through this demonstration on your own. Make
particular note of the fact that there is a unique exponential curve
passing through a given pair of points. Also note use of the
differential equation solver.

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