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BA201 ENGINEERING MATHEMATICS 2 2012
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CHAPTER 2 DIFFERENTIATION
2.1 FIRST ORDER DIFFERENTIATION
What is Differentiation?
Differentiation is all about finding rates of change of one quantity compared to another. We need differentiation when the rate of change is not constant.
Notation for the Derivative
IMPORTANT: The derivative (also called differentiation) can be written in several ways. This can cause some confusion when we first learn about differentiation.
The following are equivalent ways of writing the first derivative of y = f(x):
' ' or ( ) or dy
f x ydx
2.1.1 RULES OF DIFFERENTIATION
A. Derivative of Power Function
ny ax
So 1ndynax
dx
Examples:
1. Find the derivative of y = -7x6
Note: We can do this in one step:
We can write: OR y' = -42x5
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Example 1 Find the derivative for each of the following function.
a) 3y x b) 52y x c) xy 4 d) 2
1y
x
e) 2y x f)
2
3y x g) 3
2
3y
x h)
5 3
2y
x
i) 1
yx
j) 3
5y
x
k) y x l)
2
3
3
2
y
x
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B. Derivative of a Constant Function
y a
So 0dy
dx
Example Find the derivative for each of the following functions.
a) 1y b) y c) 40y d) 1
3y
2.1.2 THE DERIVATIVE OF SUMMATION AND SUBSTRACTION
If ( ) and ( ) are differentiable functions, the derivative of
y f x g x y f x g x
and
' 'dy
f x g xdx
' 'dy
f x g xdx
Examples:
1. Find the derivative of y = 3x5 - 1
y = 3x5 − 1
Now,
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And since we can write:
So,
2. Find the derivative of
Now, taking each term in turn:
(using )
(using )
(since -x = -(x1) and so the derivative will be -(x0) = -1)
(since )
So
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Example
Find the following derivatives;
a) 42 6y x x
b) 3 32y x x
x c) 33 4s t t d) 2 2 3p q q
e) 8 41 1
34 2
y x x
f)
4 29 5y x x x
f)
1 4 2y x
x g) 3 21 2y x x
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Exercise
Find the derivative of the following function;
i. 4 2f x x x
ii. 116 9f x x
iii. 8 5y x x
iv. 3 1y x
x
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2.1.3 THE DERIVATIVES OF COMPOSITE FUNCTION
Chain Rule
If y f u , where, u is a function of x, so:
dy dy du
dx du dx
This means we need to
1. Recognise u (always choose the inner-most expression, usually the part inside brackets, or under the square root sign).
2. Then we need to re-express y in terms of u. 3. Then we differentiate y (with respect to u), then we re-express everything in
terms of x.
4. The next step is to find du
dx.
5. Then we multiply dy
du and
du
dx.
Example 1:
Differentiate each the following function with respect to x.
i. y = (x2+ 3)5
In this case, we let u = x2 + 3 and then y = u5.
We see that:
u is a function of x and y is a function of u.
For the chain rule, we firstly need to find and .
So
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ii.
In this case, we let u = 4x2 − x and then .
Once again,
u is a function of x and y is a function of u.
Using the chain rule, we firstly need to find:
and
So
i. 3
4y x
ii. 1
4 9y
x
iii. 2 4y x
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The Extended Power Rule
An extension of the chain rule is the Power Rule for differentiating. We are finding the derivative of un (a power of a function):
k
ny ax b
1
11
kn n
kn n
dy dyk ax b ax b
dx dx
dykan ax b
dx
Example:
1.
In the case of we have a power of a function.
If we let u = 2x3 - 1 then y = u4.
So now
y is written as a power of u; and u is a function of x [ u = f(x) ].
To find the derivative of such an expression, we can use our new rule:
where u = 2x3 - 1 and n = 4.
So
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We could, of course, use the chain rule, as before: dx
du
du
dy
dx
dy*
a) 3
4y x
b) 3
5 4y x c) 2 4y x
d) 8
3 6 7y x
e)
5
12
xy
f)
1
7
xy
g) 5
1 3y
x
h)
5
2
9 4y
x
i)
7 23 24 3 2y x x x
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2.1.4 DERIVATIVE OF A PRODUCT FUNCTION
If u and v are two functions of x, then the derivative of the product uv is given by...
In words, this can be remembered as:
"The derivative of a product of two functions is the first times the derivative of the second, plus the second times the derivative of the first."
Example:
If we have a product like
y = (2x2 + 6x)(2x3 + 5x2)
we can find the derivative without multiplying out the expression on the right.
We use the substitutions u = 2x2 + 6x and v = 2x3 + 5x2.
We can then use the PRODUCT RULE:
We first find: and
Then we can write:
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Exercise:
a) 2 34 2 3 5y x x x
b) 2
1 1y x x c) 1 2y x x
d) 10 3y x x
e) 4
3 28 1y x x f) 5 6
5 1 5y x x
2.1.5 DERIVATIVE OF A QUOTIENT FUNCTION
(A quotient is just a fraction.)
If u and v are two functions of x, then the derivative of the quotient u/v is given by...
In words, this can be remembered as:
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"The derivative of a quotient equals bottom times derivative of top minus top times derivative of the bottom, divided by bottom squared."
Example:
1. We wish to find the derivative of the expression:
Solution:
We recognise that it is in the form: .
We can use the substitutions:
u = 2x3 and v = 4 − x
Using the quotient rule, we first need to find:
And
Then
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2. Find if .
Solution
We can use the substitutions:
u = 4x2 and v = x3 + 3
Using the quotient rule, we first need:
and
Then
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a)
2
353 4
xy
x
b) 28 3
2 3
xy
x
c)
432 1
1
xy
x
d) 2
1
2 3y
t
e) 2
5
2 2g s
s s
f)
22 4 3
2 3
x xg x
x
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2.1.6 DERIVATIVE OF LOGARITHMIC FUNCTION
If, lny x
1dy
dx x
lny u 1dy
dx u
lnd a
ax bdx ax b
Example:
Differentiate each of the following functions;
i. 2lny x
12
12 1
2
dy dx
dx x dx
x
x
ii. 4lny x
iii. 3
2ln 2 5y x
iv. ln 1 3y x
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Exercise:
1. 4
5lny
x
2. 2
3ln 3 7y x 3. 2ln 3 4y x x
4. ln1
xy
x
5.
4
3ln
5 3y
x
6.
5ln 1 2y x x
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2.1.7 DERIVATIVE OF EXPONENTIAL FUNCTION
If, xy e
So, xdye
dx
Example:
Differentiate each of the following functions;
i. 4xy e
4
4
4
4
4
4
x
x
x
dy de x
dx dx
e
e
ii. 23xy e
2
2
2
3 2
3
3
3
6
6
x
x
x
dy de x
dx dx
e x
xe
iii. 2
3x
y e
2
3
2
3
2
3
2
3
x
x
dy de x
dx dx
e
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Exercise:
i. 5
1 xy e ii. 3
1
2 3 xy
e
iii.
xey
x
iv. 2xy xe v. vi. 2 lnxy e x
31 xy e
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2.1.8 DERIVATIVE OF TRIGONOMETRY FUNCTIONS
If, cosdy
xdx
cosy x sindy
xdx
tany x 2secdy
xdx
Example:
Differentiate each of the following with respect to x;
i.
ii. 3cos2y x
3 sin 2 2
3 sin 2 2
6sin 2
dy dx x
dx dx
x
x
iii. tan 6y x
iv. sin 1y x
v.
vi. 3cos 3y x
vii. 52cos 2 1y x
siny x
2siny x
2cos
2cos 1
2cos
dy dx x
dx dx
x
x
2siny x
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Exercise 1:
i. 2sin 2 4y x x ii. 2
cos 2 1y x iii. tan 2y x
iv. 2
1
siny
x v. 43tan
4
xy vi. 4 1
sin2
y x
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Exercise 2:
i. cosy x x
ii. 2 sin 2y x x iii. sin
1 sin
xy
x
iv. sin 2 cos3y x x
v. tan x
yx
vi. 3 sin 2xy e x
vii. 3
3x
xy
e
viii. lny x x ix.
ln 2xy
x
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2.1.9 PARAMETRIC DIFFERENTIATION
The implicit of relationship of x and y can be expressed in a simpler form by using a third variable, known as the parameter.
Example:
Find dy
dxin terms of the parameter for
1. 2 3,x t y t t
2
2
x t
dxt
dt
3
23 1
y t t
dyt
dx
dx
dt
dt
dy
dx
dy*
t
t2
1*13 2
t
t
2
13 2
2. 22 , 4 4 4x t y t t
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3. , sintx e y t
Exercise:
Find dy
dx in terms of the parameter for
i. 33 2, 1x t y t
ii. 2 35cos , 7sinx t y t iii. 2
1, 1x y t
t
iv. 2 3 33 3 , 3x t t y t t
v. cos2 , 2 sin 2x a y a a vi. 3sin , tx t y e
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2.1.10 SECOND DERIVATIVE
The second derivative is what you get when you differentiate the derivative. Remember
that the derivative of y with respect to x is written dy
dx. The second derivative is written
2
2
d y
dx, pronounced "dee two y by d x squared".
Example:
Find dy
dx and
2
2
d y
dx if
a) 3 32y x x
x
13 32 xxx
22 316 xxdx
dy
2
2 316
xx
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Exercise:
Find dy
dx and
2
2
dx
ydif :
i. 2 345 11f t t t t
ii.2
12y
x
iii. 3 22 21 74 86y x x x
iv. 2
2 1p q
v. 35 2y x x vi. 2 1 3f x x x
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POLITEKNIK KOTA BHARU JABATAN MATEMATIK, SAINS DAN KOMPUTER
BA 201 ENGINEERING MATHEMATICS 2
PAST YEAR FINAL EXAMINATION QUESTIONS 1) Using the suitable method
differentiate the following variables.
a) ( ) ( )
b)
c)
d) ( )
e) √
2) Differentiate the equation below.
a)
b) ( )
c)
d)
e) ( )
f) ( )
3) Derive the equation below:
a) ( )( )
b) √
c) ( )
d) ( )
e)
4) Using the suitable method
differentiate the following variables
a)
b) ( )
c) ( )
d) ( )
e)
f)
( )
5) Differentiate the equation below.
a) (
)
b) √
c)
d)
e)
6) Derive the equation below:
a)
b) ( )
c) ( )
d)
e)
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7) Find
for the following equations
a)
b) ( )( )
c)
d)
e) ( ) ( )
f)
√
8) Find
for the following equations
a)
b) ( )
c) √
d)
e)
9) Using the right method, differentiate
the functions given.
a) (
)
b) ( )
c)
√
d) ( )
e)
10) Derive the equation below:
a) √
b)
c)
d) ( )( )
e) ( )
f) ( )
11) Find the dy
dxfor the parametric
functions given below in terms of t.
a) 4 333 5,
4y t x t t
b) 2 25 , lny t x t
c) 2 32 3, 4x t y t t
12) Find the second derivatives for the
function
a) 2
5 2 3 1y x x
b) 3
3 5f x x
c) 2
24 2 3y x x