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

2. Functions of the type
Recall our earlier discussions of the porosity depth relationship

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

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

5. As we converge toward 1km, /z decreases to -0
As we converge toward 1km, /z decreases to km-1 between 1 and 1.1 km depths.

6. What is the gradient at 1km?

7. Computer evaluation of the derivative

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

10. Remember the idea of the dy and dx is that they represent differential changes that are infinitesimal - very small.
So if dx is (that’s 1x10-4) then (dx)2 = (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

11. so that we can just ignore all those terms with (dx)n where n is greater than 1.
Our equation gets simple fast
Also, since y =x4, we have
and then -

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

13. 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 -

14. Examining the effects of differential increments in y and x we get the following

15. Don’t let negative exponents fool you
Don’t let negative exponents fool you. If n is -1, for example, we still have
or just

16. Take a couple minutes to
evaluate the power rule examples
on your worksheet

17. The sum rule - Given the function - what is ?
We just differentiate f and g individually and take their sum, so that
Show ths works for y + y, = f + df + g+dg … divide through by dx

18. Take the simple example
- what is ?
What are the individual derivatives of
and ?

19. We know from the forgoing that the c disappears.
let
then -
We just apply the power rule and obtain
We know from the forgoing that the c disappears.

20. We use the power rule again to evaluate the second term, letting
g = (ax4+b)
Thus -

21. Differences are treated just like sums
so that
is just

22. evaluate the examples on your worksheet

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

24. Since dfdg is very small and since y=fg, the above becomes -
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 -

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

26. 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

27. 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

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

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

30. Since
and
in general
a can be thought of as a general base. It could be 10 or 2, etc.

31. The derivative of logarithmic functions
Given >
We’ll talk more about these special cases after we talk about the chain rule.

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

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

34. 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

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

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

37. In general if
then

38. 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

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

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

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

42. the ln and exponential rules
Use the chain rule,
the ln and exponential rules
to differentiate the
examples in the handout

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

44. Finish reading Chapter 8
Differential Calculus