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Math 2413
Notes 2.2
Section 2.2–Derivatives of Polynomials and Trigonometric Functions
Recall definition of derivative: lim
h0
f ( a  h)  f ( a )
if the limit exists.
h
However, this can be very difficult for some functions. We now have some rules that will make the
computations much easier.
Constant rule: If f (x) = k, where k is a real number, then f ' (x) = 0 .
Power Rule: If f (x) = xn, where n is any real number not equal to 0, then f ' (x) = nx n-1.
Example 1: Find derivative of the following functions:
1.
f ( x)  11
2.
f (x )  
3.
f ( x)  x
4.
f ( x)  x 4
5.
f ( x)  x
6.
f ( x) 
1
x7
1
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Math 2413
Notes 2.2
Theorem 2.2.1 : Let k be a real number. If f and g are differentiable at x, then so are f + g ,
f − g and k ⋅ f . Moreover,
• (f ± g)' (x) = f ' (x) ± g' (x),
• (k ⋅ f)' (x) = k ⋅ f ' (x).
That is, the derivative of the sum of two functions is the sum of the derivatives. And, the derivative of a scalar
multiple of a function is the scalar multiple of the derivative.
Theorem 2.2.1 can be extended to any collection of finitely many functions:
Fact: For any linear combination of functions f1, f2 ,..., fn;
(k1 f1 + k2 +...+ kn fn )' = k1 f1'+ k2 f2'+...+ kn fn'
Using this fact and the power rule, we can compute the derivative of any polynomial.
Theorem 2.2.2: Derivatives of Polynomials
Let f ( x )  an x n  an1 x n1  ........  a2 x 2  a1 x  a0 f (x) be a polynomial function. The derivative is:
f ' ( x)  an  nx n1  an1 (n  1) x n2  ........  a2  2 x  a1
Note that since the derivative of a constant number is 0, the constant term of a polynomial disappears while
differentiating the function.
Example 2: Find the derivative of each function:
1.
f ( x)  5 x 4
2.
f ( x) 
7
x3
2
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Math 2413
Notes 2.2
3.
f ( x)  4 x 5  6 x 4  x 2  9 x  17
4.
f ( x)  3 x 
2
 8x
x
Example 3: Find the slope of the line that is tangent to the graph of f ( x )  x 3  4 x 2  8 x  6 at
the point x = 2 .
3
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Math 2413
Notes 2.2
Since tangent and normal lines are perpendicular, the slope of the normal line is the negative (or opposite)
reciprocal of the slope of tangent:
1
1
mnormal  

mtangent
f ' (a )
Example 4: Find the slope and the equation of the tangent line to the graph of the function at the specified
5
5

point. f ( x )   x 2  2 x  2 at   1, 
3
3

Example 5: Let f(x) = x3 – 4x2. Find the point(s) on the graph of f where the tangent line is horizontal.
4
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Math 2413
Notes 2.2
Example 6: Find the slope and the equation of the normal line to the graph of the function f(x) = 2x 2 – 3x + 4 at
(2, 6).
Example 7: Given the function f(x) = 3x3 + 2x2 – 8x + 1, find the point(s) where the tangent line has a slope of
5.
5
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Math 2413
Notes 2.2
Derivatives of the Six Trigonometric Functions:
(sin x)' = cos x
(cos x)' = - sin x
(tan x)' = sec2 x
(cot x)' = - csc2 x
(sec x)' = sec x tanx
(csc x)' = - csc x cot x
Example 8: Given function f ( x)  tan x , write an equation of the tangent line at the point (c, f (c)) where

c .
4
Example 9: Given function f ( x )  cos x , write an equation of the tangent line at the point (c, f (c)) where

c .
3
6
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Math 2413
Notes 2.2
where c   .
Example 10: Find f ' (c) if f ( x)  2 x  3 cos x  tan x
Example 11: Consider function f (x) = sin x over the interval [0, 2π ] ; find all point(s) on the graph of f where
the slope of the normal line is −2.
Example 12: Consider function f ( x )  x 4  8 x 3  13 , find all point(s) on the graph of f where the tangent line is
horizontal.
7
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Math 2413
Notes 2.2
Leibniz’s d/dx Notation
So far, we have used the “prime” notation for the derivative. There are many other notations that are widely
used. One of the most commonly used notations is Leibniz’s “double-d” notation:
The derivative of a function y can be denoted as
dy
if y is a function in terms of x ,
dx
dy
if y is a function in terms of t ,
dt
and so on.
Here,
dy
indicates the derivative of y with respect to x.
dx
Example 12: Find the following:
a.
dy
;
dx
b.
dy
2 x  5
dx
y  x3  sin x
Higher Order Derivatives
Sometimes we need to find the derivative of the derivative. Since the derivative is a function, this is something
we can readily do. The derivative of the derivative is called the second derivative, and is denoted f ''(x).
Similarly, the third derivative is the derivative of the second derivative, the fourth derivative is the derivative of
the third derivative, the fifth derivative is the derivative of the fourth derivative, etc.
The second, third, fourth, fifth, . . . derivatives of a function are collectively called higher order derivatives.
8
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Math 2413
Notes 2.2
Notations:
If y  f (x) , the function f ' , f " , f ' ' ' , f ( 4) ,...., f ( n ) are called the derivatives of f of order 1, 2, 3, 4, …, n
respectively.
First derivative:
y '  f ' ( x) 
dy
dx
Second derivative: y"  f " ( x) 
Third derivative:
or
y'  f ' ( x) 
d 2 y d  dy 
  
dx 2 dx  dx 
y" '  f " ' ( x) 
d
 f ( x) 
dx
y"  f " ( x ) 
or
d3y d  d2y 
 or
 
dx 3 dx  dx 2 
d2
 f ( x)  d  d  f ( x)
2
dx
dx  dx

y" '  f " ' ( x ) 
And so on.
 
Example 13: Given f ( x)  sin x  3 cos x  4 . Find f "   .
2
d3y
Example 14: Given f ( x)  x  5 x  3 . Find
.
dx3
4
2
9
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
d3y d  d2

 2  f ( x)
3
dx
dx  dx
