Geology 351 - GeoMath
Derivatives (p2)
tom.h.wilson
tom.wilson@mail.wvu.edu
Dept. Geology and Geography
West Virginia University
Tom Wilson, Department of Geology and Geography
Objectives for the day
•
•
•
•
•
Reminders
Product and quotient rules
General comments about trigonometric functions
Wrap up derivatives worksheet
Hand out some additional examples for practice
Tom Wilson, Department of Geology and Geography
At 2.5km, the slope is -0.05
at 3.5 km the slope is -0.0259
at 0.5, -0.191
Porosity depth relationship
0.4
-z/1.5
=0.4e
0.3
PHI
0.2
0.1
0.0
-0.1
-0.2
-0.3
0
1
2
3
Z
With slope varying as
Tom Wilson, Department of Geology and Geography
4
d 1

0.4e  z /1.5
dz 1.5
5
0, -0.707 (at 0.785), -1 (at 1.571), -0.707 (at 2.356)
Cosine function
1.0
cos ()
0.5
0.0
-0.5
 (radians)
Tom Wilson, Department of Geology and Geography
0
6.
5
5.
0
5.
5
4.
0
4.
5
3.
0
3.
5
2.
0
2.
5
1.
0
1.
5
0.
0.
0
-1.0
For y=x2
2
y=x
40
35
30
25
y 20
Tangent line
15
10
5
0
0
1
2
3
x
4
Slope of the
5
6line
tangent
Slope or derivative = 2x
Tom Wilson, Department of Geology and Geography
Product and quotient rules
How do you handle derivatives of functions like
y ( x)  f ( x) g ( x)
or
f ( x)
y ( x) 
?
g ( x)
The products and quotients of other
functions
Tom Wilson, Department of Geology and Geography
Removing explicit reference to the
independent variable x, we have
y  fg
Going back to first principles, we have
y  dy  ( f  df )( g  dg)
Evaluating this yields
y  dy  fg  gdf  fdg  dfdg
Since dfdg is very small we let it equal
zero; and since y=fg, the above becomes
Tom Wilson, Department of Geology and Geography
dy  gdf  fdg
Which (after division by x) 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
f
y
g
Tom Wilson, Department of Geology and Geography
The quotient rule states that
d f
  
dx  g 
g df
dx
 f dg
dx
g2
The proof of this relationship can be
tedious, but I think you can get it much
easier using the power rule
Rewrite the quotient as a product and apply the
product rule to y as shown below
f
y   fg 1
g
Tom Wilson, Department of Geology and Geography
We could let h=g-1 and then rewrite y as
y  fh
Its derivative using the
product rule is just
dy
df
dh
h  f
dx
dx
dx
dh = -g-2dg and substitution yields
dy

dx
Tom Wilson, Department of Geology and Geography
df
dx 
g
f dg
g2
dx
Multiply the first term in the sum by g/g (i.e. 1) to get >
dy g

dx g
df
dx 
g
f dg
dx
g2
Which reduces to
dy

dx
g df
dx
 f dg
dx
g2
the quotient rule
Tom Wilson, Department of Geology and Geography
Tom Wilson, Department of Geology and Geography
Hand in before leaving
Finish up the in-class
problems
Tom Wilson, Department of Geology and Geography
Look over problems 8.13 and 8.14
•Bring questions to class this Thursday
•Due date – Tuesday, March 25th
Tom Wilson, Department of Geology and Geography
Next time we’ll continue with exponentials and logs, but also
have a look at question 8.8 in Waltham (see page 148).
Find the derivatives of
(i )   x 2 .e x
(ii) y  3 2 .sin(  )
(iii) z  x.cos(x)  x . tan( x)
2
(iv) B  3 4 . ln( )  17 2
Tom Wilson, Department of Geology and Geography
Due dates ….
Spend some time going over the graphical analysis of
slopes for the porosity- depth relationship, cosine
and x2 functions.
Wrap up the basic in-class differentiation worksheet
and hand in before leaving
Next time we’ll talk about derivatives of logs and
exponential functions and use Excel to compute the
derivative of exponential functions for in-class
illustration and discussion
• continue reading Chapter 8 – Differential Calculus
Tom Wilson, Department of Geology and Geography
Next time we’ll talk about differentiation of log
and exponential functions
We’ll use the computer this time to help us
conceptualize the derivative or slope
x
de
 ex
dx
cx
dAe
cx d (cx )
 Ae
 cAecx
dx
dx
This is an application of the rule for differentiating
exponents and the chain rule
Tom Wilson, Department of Geology and Geography
Exponential functions in the form Ae
 cx
Porosity-Depth Relationship
0.5
See chapter 8
0.4
0.3

0.2
0.1
Slope
0.0
0
1
2
3
4
5
Z (km)
Recall our earlier discussions of the
porosity depth relationship
Tom Wilson, Department of Geology and Geography
  o e
 cz
  o e
 cz
Derivative concepts
Porosity-Depth Relationship
0.5
0.4
0.3

0.2
0.1
Slope
0.0
0

?
z
Tom Wilson, Department of Geology and Geography
1
2
3
4
5
Z (km)
Refer to graphical exercise and to
comments on the computer lab exercise.
  o e  cz
Porosity-Depth Relationship
0.5
0.4
0.3

0.2
0.1
Slope
0.0
0
1
2
3
4
5
Z (km)

?
z
Tom Wilson, Department of Geology and Geography
Between 1 and 2 kilometers the
gradient (slope) is -0.12 km-1
In the limit that our computations converge on a point
we have the slope (derivative) at that point.
Porosity-Depth Relationship
0.34
0.32
0.30
Gradient
1 to 2 km
0.28

0.26
Gradient
1.0 to 1.1 km
0.24
0.22
0.20
0.18
0.16
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
Z (km)
As we converge toward 1km, /z decreases to -0.14
km-1 between 1 and 1.1 km depths.
Tom Wilson, Department of Geology and Geography
Porosity-Depth Relationship
0.34
0.32
0.30
Gradient
1 to 2 km
0.28

0.26
Gradient
1.0 to 1.1 km
0.24
0.22
0.20
0.18
0.16
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
Z (km)
We’ll talk more about this and logs next time
What is
Tom Wilson, Department of Geology and Geography
d ?
dz
Due dates ….
Spend some time going over the graphical analysis of
slopes for the porosity- depth relationship, cosine
and x2 functions.
Wrap up the basic in-class differentiation worksheet
and hand in before leaving
Next time we’ll talk about derivatives of logs and
exponential functions and use Excel to compute the
derivative of exponential functions for in-class
illustration and discussion
• continue reading Chapter 8 – Differential Calculus
Tom Wilson, Department of Geology and Geography