Lesson 5-Differentiation of Algebraic Functions
Lesson 5-Differentiation of Algebraic Functions
Differentiation of Algebraic
Functions
OBJECTIVES:
• to identify the different rules of differentiation and distinguish one
from the other;
• to prove the different rules of differentiation using the increment
method;
• to find the derivative of an algebraic function using the basic rules of
differentiation; and
• to extend these basic rules to other “complex” algebraic functions.
DERIVATIVE USING
FORMULAS
6
dy 6 1
x 7
dx 7
dy 6 67 77 6 17 6 67 x 6
x x 7
dx 7 7 7 x 7x
Derivative of a Constant Times a
Function
Theorem: ( Constant Multiple Rule) If f is
a differentiable function at x and
c is any real number, then cf is
also differentiable at x and
d d
cf ( x) c f ( x)
dx dx
In words, the derivative of a constant times a
function is the constant times the derivative of
the function, if this derivative exists.
Proof:
d cf ( x x) cf ( x)
cf ( x) lim
dx x 0 x
f ( x x) f ( x)
lim c
x 0
x
f ( x x) f ( x)
c lim
x 0 x
d
c f (x)
dx
Example : Differenti ate the following functions
1. y 5x 8 2. f(x) 9 x 4
y' 5 8 x 7
f' (x) 9 - 4 x 4 1
5 36
y' 40 x 7
f' (x) 36 x 5
x
Derivatives of Sums or Differences
Theorem: ( Sum or Difference Rule)f If f and g are both
differentiable functions at x, then so are and f g
and
f g
d d d d
f g d f d g or f ( x ) g ( x ) f ( x ) g ( x )
dx dx dx dx dx dx
f ( x x) f ( x) g ( x x) g ( x)
lim lim
x 0 x x 0 x
d d
[ f ( x)] [ g ( x)]
dx dx
Proof:
d [ f ( x x) g ( x x) [ f ( x) g ( x)]
[ f ( x) g ( x)] lim
dx x 0 x
[ f ( x x) f ( x)] [ g ( x x) g ( x)]
lim
x 0 x
f ( x x) f ( x) g ( x x) g ( x)
lim lim
x 0 x x 0 x
d d
[ f ( x)] [ g ( x)]
dx dx
Example : Differenti ate the following functions
1. y 5x 4 6 x 2 4x 7 2. f(x) 2 x 4 9 x 4
y' 20 x 3 12 x 4 f' (x) 8 x 5 9
y' 4 5x 3x 1
-8
3
f' (x) 5 9
x
Derivative of a Product
Theorem: (The Product Rule) If f and g are
both differentiable functions at x, then so is the
product f ● g, and
d dg df
f g f g or
dx dx dx
d d d
f ( x) g ( x) f ( x) [ g ( x)] g ( x) f ( x)
dx dx dx
In words the derivative of a product of two functions is the first
function times the derivative of the second plus the second
function times the derivative of the first, if these derivatives exist.
Proof:
d f ( x x) g ( x x) f ( x) g ( x)
[ f ( x) g ( x)] lim
dx x 0 x
f ( x x) g ( x x) f ( x x) g ( x) f ( x x) g ( x) f ( x) g ( x)]
lim
x 0 x
g ( x x) g ( x) f ( x x) f ( x)
lim f ( x x) g ( x)
x 0 x x
g ( x x) g ( x) f ( x x) f ( x)
lim f ( x x) lim lim g ( x) lim
x 0 x 0 x x 0 x 0 x
d d
lim f ( x x g ( x) lim g ( x) f ( x)
x 0 dx x 0 dx
Example : Differenti ate the following functions and simplify.
1. y 3x 4 4x 2 3
y' 3x 4 8x 4x 2 3 3
y' 24x 2 32x 12x 2 - 9
y' 36x 2 32x - 9
2. y x 3 1 5 - 2x
y' x 3 1 - 2 5 - 2x 3x 2
y' -2x 3 2 15x 2 - 6x 3
y' -8x 3 15x 2 2
Derivative of a Quotient
Theorem: (The Quotient Rule) If f and g are
both differentiable functions at x, and if g(x) ≠ 0
then f/g is differentiable at x and
df dg
g f
d f
dx 2 dx or
dx g g
d d
g ( x) f ( x) f ( x) g ( x)
d f ( x) dx dx
dx g ( x) g ( x ) 2
In words, the derivative of a quotient of two functions is
the fraction whose numerator is the denominator times
the derivative of the numerator minus the numerator
times the derivative of the denominator and whose
denominator is the square of the given denominator
Example : Differenti ate the given function and simplify.
4x 2 3
y
1 2x
1 2x 8x 4x 2 3 2
y'
1 2x 2
8x 16x 2 8x 2 6
y'
1 2x 2
8x 2 8x 6
y'
1 2x 2
2 4x 2 4x 3
y'
1 2x 2
Derivatives of Composition
Theorem: (The Chain Rule) If g is differentiable at x and
if f is differentiable at g(x), then the composition f ◦ g is
differentiable at x. Moreover, if y=f(g(x)) and u=g(x) then
y=f(u) and
dy dy du
dx du dx
or
du n
nu n 1 du
dx dx
Example : Differenti ate the following functions and simplify.
2
1. y 3x 10x 15 5
2
y' 5 3x 10x 15 6x - 10
4
3
5
2. G(x)
x -1
2
5 5 125
G'(x) 3
x -1 x 1 2 3 x 1 4
3. y 3x 1 4x 5 4
y 4 3x 1 4x 5 3 3x 1 4 4x 5 3
y 4 3x 1 4x 5 3 12x 4 12x - 15
y 4 24x - 11 3x 1 4x 5 3
Derivative of a Radical with index equal to 2
If u is a differentiable function of x, then
du
d
dx
u dx
2 u
The derivative of a radical whose index is
two, is a fraction whose numerator is the
derivative of the radicand, and whose
denominator is twice the given radical, if the
derivative exists.
Derivative of a Radical with index other than 2
If n is any positive integer and u is a differentiable function of
x, then
d 1
1
1 n 1 du
u u
n
dx n dx
2. y 5 2x 4 x 5
1
y 2x 4 x 5 5
1 1
y' 2x 4 x 5 5 1 2x 4 1 x 5 2
5
1 4
y' 2x 4 x 5 5 2x 4 2x 10
5
1 4
y' 2x 4 x 5 5 4x 14
5
1 4
y' 2x 4 x 5 5 2x 4 2x 10
5
1 4
y' 2x 4 x 5 5 4x 14
5
2 2x 7
y' 4
5 2x 4 x 5 5