Download Calculus 1 Lecture Notes

Calculus 1 Lecture Notes
Section 2.3
Page 1 of 5
Section 2.3: Computation of Derivatives: The Power Rule
Map of Chapter 2: Differentiation
1. What is differentiation?
a. It is the process of finding the slope of the tangent line to the graph of a function at a
point. This slope is found by taking the limit of the slopes of secant lines that get closer
f  x  h  f  x
and closer to the point in question. (Sections 2.1 and 2.2) f '( x)  lim
h 0
h
2. What are some shortcuts for calculating derivatives? Group the shortcuts by the functions
we know about:
a. Constant and linear functions (Section 2.3)
b. Power functions (The Power Rule Section 2.3)
c. Polynomial functions (Consequence of Section 2.3)
d. Trigonometric functions (Section 2.5)
e. Exponential and Logarithmic Functions (Section 2.6)
f. Combinations of functions:
i. Linear combinations; i.e., adding functions together (Section 2.3)
ii. Products of functions (The Product Rule Section 2.4)
iii. Quotients of functions; rational functions (The Quotient Rule Section 2.4)
iv. Compositions of functions (The Chain Rule Section 2.7)
3. What is some derivative stuff that I would never think of?
a. Differentiability implies continuity (Section 2.2)
b. Implicit differentiation (Section 2.8)
c. The Mean Value Theorem (Section 2.9)
Big Idea for Section 2.3: A shortcut for calculating the derivative of a power function is that the
derivative is equal to the exponent times the variable raised to the power of one minus the original
d r
x  rx r 1 .
exponent:
dx
Calculus 1 Lecture Notes
Derivative of f(x) = x
f  x  h  f  x
f '( x)  lim
h 0
h
 x  h   x
 lim
h 0
h
h
 lim
h 0 h
1
 1 x0
Section 2.3
Derivative of g(x) = x2
g  x  h  g  x
g '( x)  lim
h 0
h
 x  h  2    x 2 
  
 lim 
h 0
h
 x 2  2 xh  h 2    x 2 
 lim 
h 0
h
2
2 xh  h
 lim
h 0
h
 lim  2 x  h 
h 0
Page 2 of 5
Derivative of h(x) = x3
h  x  h  h  x
h '( x)  lim
h 0
h
  x  h 3    x 3 
  
 lim 
h 0
h
 x3  3 x 2 h  3 xh 2  h3    x 3 
 lim 
h 0
h
2
2
3 x h  3xh  h3
 lim
h 0
h
2
 lim  3 x  3 xh  h 2 
 2x
 2 x1
What seems to be the pattern of the answers?
How would you write that pattern mathematically?
Theorem 3.1: Derivative of a constant function
d
c  0 for any constant c
dx
Proof: Let f(x) = c.
d
f x  h   f x 
cc
0
c  f ' ( x)  lim
 lim
 lim  0
h

0
h

0
h

0
dx
h
h
h
Theorem 3.2: Derivative of f(x) = x
d
x 1
dx
Proof: Let f(x) = x.
x  h   x   lim h  1
d
f x  h   f x 
x  f ' ( x)  lim
 lim
h 0
h 0
h 0 h
dx
h
h
h 0
 3x 2
Calculus 1 Lecture Notes
Section 2.3
Page 3 of 5
Theorem 3.3: (Power Rule) Derivative of a positive integer power function
d n
x  nx n 1 for any integer n > 0
dx
Proof: Let f(x) = xn.
f  x  h  f  x
d n
x  f '( x)  lim
h 0
dx
h
 x  h
 lim
h 0
n
 xn
h
n(n  1) n  2 2
x h  ...  nxh n 1  h n  x n
2
 lim
h 0
h
n(n  1) n  2


h  nx n 1 
x h  ...  nxh n  2  h n 1 
2

 lim 
h 0
h
n(n  1) n  2


 lim  nx n 1 
x h  ...  nxh n  2  h n 1 
h 0
2


n 1
 nx
x n  nx n 1h 
Theorem 3.4: (General Power Rule) Derivative of any power function
d r
x  rx r 1 for any real number r.
dx
Proof: see http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/proofs/powerruleproof.html
for a nice, readable proof that handles all the cases: r a positive integer, r a negative integer, r a
rational number, and r an irrational number.
Theorem 3.4: Derivatives of Linear Combinations of Functions
d
 f x   g x   d f x   d g x 
1. The derivative of a sum is the sum of the derivatives:
dx
dx
dx
Proof: Let k(x) = f(x) + g(x).
d
k x  h   k x 
 f x   g x   k ' ( x)  lim
h 0
dx
h
 f x  h   g x  h    f x   g x 
 lim
h 0
h
 f x  h   f x   g x  h   g x 
 lim
h 0
h
 f x  h   f x   lim g x  h   g x 
 lim
h 0
h 0
h
h
 f ' x   g ' x 
Calculus 1 Lecture Notes
Section 2.3
Page 4 of 5
2. The derivative of a difference is the difference of the derivatives:
d
 f x   g x   d f x   d g x 
dx
dx
dx
Proof: Let k(x) = f(x) - g(x).
d
k x  h   k x 
 f x   g x   k ' ( x)  lim
h 0
dx
h
 f x  h   g x  h    f x   g x 
 lim
h 0
h
 f x  h   f x   g x  h   g x 
 lim
h 0
h
 f x  h   f x   lim g x  h   g x 
 lim
h 0
h 0
h
h
 f ' x   g ' x 
3. The derivative of a product with a constant is the product of the constant and the
d
cf x   c d f x 
derivative:
dx
dx
Proof: Let k(x) = cf(x).
k  x  h  k  x
d
cf  x    k '( x)  lim
h 0
dx
h
cf  x  h   cf  x 
h
 f  x  h  f  x 
 lim c 

h 0
h


f  x  h  f  x
 c lim
h 0
h
 cf '  x 
 lim
h 0
Higher order derivatives: (derivative of a derivative):
df
1. First order derivative: y '  f '( x) 
 x
dx
d2 f
2. Second order derivative: y ''  f ''( x)  2  x 
dx
d3 f
3. Third order derivative: y '''  f '''( x)  3  x 
dx
d4 f
(4)
(4)
4. Fourth order derivative: y  f ( x)  4  x 
dx
Calculus 1 Lecture Notes
Section 2.3
Page 5 of 5
Physics terminology:
The first derivative of the position function is called instantaneous velocity.
The second derivative of the position function is called instantaneous acceleration.
The third derivative of the position function is called the “jerk”.
Practice
1. Calculate the derivative of f  x   x3  x
2. What is the equation of the tangent line to f(x) at x = 5?
d  3
3x  2 x  4 x  1 
3.
dx 
d
d
d
d
d
3x 3  2 x  4 x  1  3x 3  2 x  4 x  1
dx
dx
dx
dx
dx
d
 33 x 2   21  4 x 1 / 2  0
dx



1
 9 x 2  2  4 12 x 2
 9x 2  2  2x
 12
1

 9x 2  2 
2
x
4. Calculate the derivative of g  x   3 x 2
5. Calculate the derivative of f  t   5t e  3t 0.9
6. Calculate the derivative of f  x    x  1 x  5
7. Calculate the derivative of f  x  
x2  2 x  4
2x
8. Calculate f (x) for f(x) = x2
9. Calculate f '''( x) for f  x   x 2  x