1. The exponential function occurs very frequently in mathematical models of nature and society.

2. 11.71828 0.11.05171 0.011.00502 0.0011.00050 0.00011.00005 0.000011.00001 So, how to calculate the slope when you don’t know derivative?

3. example: Differentiate the function example:

4. take derivative of both sides take ln of both sides ?

6. Differentiate y = ln(x 3 + 1). example: To use the Chain Rule, we let u = x 3 + 1. Then, y = ln u. example: Find:

7. Differentiate example:

8. If we first simplify the given function using the laws of logarithms, the differentiation becomes easier

9. example: Thus, f ’(x) = 1/x for all x ≠ 0. The result is worth remembering:

10. a logarithmic function with base a in terms of the natural logarithmic function: example:

11. LOGARITHMIC DIFFERENTIATION 1.Take natural logarithms of both sides of an equation y = f(x) and simplify. 2. Differentiate implicitly with respect to x. 3. Solve the resulting equation for y’. example: differentiate Since we have an explicit expression for y, we can substitute and write If we hadn’t used logarithmic differentiation the resulting calculation would have been horrendous.

12. example: