Geomath

1. Geomath Geology 351 - Differential Calculus tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

2. Derivative concepts Refer to comments on the computer lab exercise.

3. Between 1 and 2 kilometers the gradient is -0.12 km-1

4. As we converge toward 1km, /z decreases to -0.14 km-1 between 1 and 1.1 km depths.

5. What is the gradient at 1km? What is ?

6. Computer evaluation of the derivative

7. The power rule - The book works through the differentiation of y = x2, so let’s try y =x4. multiplying that out -- you get ...

8. Remember the idea of the dy and dx is that they represent differential changes that are infinitesimal - very small. So if dx is 0.0001 (that’s 1x10-4) then (dx)2 = 0.00000001 (or 1x10-8) (dx)3 = 1x10-12 and (dx)4 = 1x10-16. So even though dx is very small, (dx)2 is orders of magnitude smaller

9. Divide both sides of this equation by dx to get This is just another illustration of what you already know as the power rule,

10. which - in general for is Just as a footnote, remember that the constant factors in an expression carry through the differentiation. This is obvious when we consider the derivative -

11. The sum rule - Given the function - what is ? We just differentiate f and g individually and take their sum, so that

12. Product and quotient rules - Recall how to handle derivatives of functions like or ?

13. Removing explicit reference to the independent variable x, we have Going back to first principles, we have Evaluating this yields Since dfdg is very small and since y=fg, the above becomes -

14. The quotient rule is just a variant of the product rule, which is used to differentiate functions like Which is a general statement of the rule used to evaluate the derivative of a product of functions.

15. The Chain Rule - Differentiating functions of functions - Given a function we consider write compute Then compute and take the product of the two, yielding

16. Outside to inside rule We can also think of the application of the chain rule especially when powers are involved as working form the outside to inside of a function

17. Where Again use power rule to differentiate the inside term(s) Derivative of the quantity squared viewed from the outside.

18. Using a trig function such as let then Which reduces to or just

19. In general if then

20. Returning to those exponential and natural log cases - we already implemented the chain rule when differentiating h in this case would be ax and, from the chain rule, becomes or and finally since and

21. For functions like we follow the same procedure. From the chain rule we have Let and then hence

22. Thus for that porosity depth relationship we were working with -

23. For logarithmic functions like We combine two rules, the special rule for natural logs and the chain rule. Log rule then Let Chain rule and so

24. The quotient rule states that And in most texts the proof of this relationship is a rather tedious one. The quotient rule is easily demonstrated however, by rewriting the quotient as a product and applying the product rule. Consider

25. We could let h=g-1 and then rewrite y as Its derivative using the product rule is just dh = -g-2dg and substitution yields

26. Multiply the first term in the sum by g/g (i.e. 1) to get > Which reduces to i.e. the quotient rule

27. Special Cases- Given > • The derivative of an exponential function In general for If express a as en so that then Note

28. Take a few moments to through the examples on your worksheet

29. For next time look over question 8.8 in Waltham (see page 148). Find the derivatives of

30. Finish reading Chapter 8 Differential Calculus