3.9: Derivatives of Exponential and Logarithmic Functions

Definition of e: or also The essential point for both is that the exponent is the reciprocal of the number added to 1. reciprocals

Evaluate: Solution: Let 2h = t, then and t→0 when h→0 = e 2

Do Harcourt Calculus Text problems from Section 6.2

Look at the graph of The slope at x=0 appears to be 1. If we assume this to be true, then: definition of derivative

Now we attempt to find a general formula for the derivative of using the definition. This is the slope at x=0, which we have assumed to be 1.

is its own derivative! The geometric significance of this is that the slope of the tangent to the curve y = e x is equal to the y-coord. of the point. Also, y = e x is the only function of the form y = b x that crosses the y-axis with a slope of 1.

Find the maximum and minimum of f(x) = xe -2x over the interval 0 < x < ln3. Solution: f’(x) = e -2x -2xe -2x = e -2x (1-2x) For Critical points: f’(x) = 0 when x = 0.5 (since e -2x > 0 for all x) x f(x) Maximum point ln3 Minimum point

If we incorporate the chain rule: We can now use this formula to find the derivative of is its own derivative!

( and are inverse functions.) (chain rule)

( is a constant.) Incorporating the chain rule:

Natural Logarithms Recall log e x = lnx Then ln x = y ↔ e y = x Also worth noting are ln e = 1 ln 1 = 0

So far we have: Now it is relatively easy to find the derivative of.

To find the derivative of a common log function, you could just use the change of base rule for logs: The formula for the derivative of a log of any base other than e is:

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