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

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

Derivative concepts Refer to comments on the computer lab exercise.

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

As we converge toward 1km, /z decreases to -0 As we converge toward 1km, /z decreases to -0.14 km-1 between 1 and 1.1 km depths.

What is the gradient at 1km?

Computer evaluation of the derivative

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

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

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 -

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

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 -

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

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

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

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

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

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.

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

Differences are treated just like sums so that is just

evaluate the examples on your worksheet

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

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 -

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

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

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

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

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

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

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

Take a few moments to through the examples on your worksheet

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

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

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

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

In general if then

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

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

Thus for that porosity depth relationship we were working with -

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

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

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

Finish reading Chapter 8 Differential Calculus