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ENGINEERING MATHEMATICS 2 BA201
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):
2.1.1 RULES OF DIFFERENTIATION
A. Derivative of Power Function
So
Examples:
1. Find the derivative of y = -7x6
Note: We can do this in one step:
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We can write: OR y' = -42x5
Example 1 Find the derivative for each of the following function.
a) b) c) y=4√ xd)
e) f) g) h)
i) j)
k)
l)
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B. Derivative of a Constant Function
So
Example Find the derivative for each of the following functions.
a) b) c)d)
2.1.2 THE DERIVATIVE OF SUMMATION AND SUBSTRACTION
If f ( x ) and g ( x ) are differentiable functions, the derivative of
and
Examples:
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1. Find the derivative of y = 3x5 - 1
y = 3x5 − 1
Now,
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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ExampleFind the following derivatives;
a)b)
c)d)
e)
f) g)
h)
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Exercise
Find the derivative of the following function;
i.
ii.
iii.
iv.
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2.1.3 THE DERIVATIVES OF COMPOSITE FUNCTION
Chain Rule
If , where, u is a function of x, so:
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 .
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5. Then we multiply and .
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
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
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So
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):
Example:
1.
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i.ii.
iii.
ENGINEERING MATHEMATICS 2 BA201
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
We could, of course, use the chain rule, as before: dydx
=dydu
∗dudx
a) b) c)
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d)
e)f)y=− 7
√ x+1
g) h)i)
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:
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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:
Exercise:
a) b) c)
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d) e) f)
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:
"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:
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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
2. Find if .
Solution
We can use the substitutions:
u = 4x2 and v = x3 + 3
Using the quotient rule, we first need:
and
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Then
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a) b) c)
d) e) f)
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Challenge
Find the derivative of
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2.1.6 DERIVATIVE OF LOGARITHMIC FUNCTION
If,
Example:
Differentiate each of the following functions;
i.
ii.
iii.
iv.
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Exercise:
2.1.7 DERIVATIVE OF EXPONENTIAL FUNCTION
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1. 2. 3.
4. 5.6.
ENGINEERING MATHEMATICS 2 BA201
If,
So,
Example:
Differentiate each of the following functions;
i.
ii.
iii.
Exercise:
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i. ii. iii.
iv. v.31 xy e vi.
2.1.8 DERIVATIVE OF TRIGONOMETRY FUNCTIONS
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If, siny x
Example:
Differentiate each of the following with respect to x;
i. 2siny x
2cos
2cos 12cos
dy dx xdx dx
xx
ii.
iii.
iv.
v.2siny x
vi.
vii.
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Exercise 1:
i. ii. iii.
iv. v. vi.
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Exercise 2:
i. ii.iii.
iv.v.
vi.
vii.viii.
ix.
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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 in terms of the parameter for
1.
dydx
=dydt
∗ dtdx
=3 t2+1∗ 1
2 t
=3 t
2+12 t
2.
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3.
Exercise:
Find in terms of the parameter for
i. ii.iii.
iv. v. vi.
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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 . The second derivative is written
, pronounced "dee two y by d x squared".
Example:
Find and if
a)
=2 x3+x−3 x−1
dydx
=6 x2+1+3 x−2
=6 x2+1+ 3
x2
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Exercise:
Find and
d2 ydx2 if :
i.ii.
iii.
iv. v. vi.
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POLITEKNIK KOTA BHARUJABATAN MATEMATIK, SAINS DAN KOMPUTER
BA 201 ENGINEERING MATHEMATICS 2
PAST YEAR FINAL EXAMINATION QUESTIONS
1) Using the suitable method
differentiate the following variables.
a) y= (x+1 )2 ( x−1 )6
b) y= cos2 xsin 3 x
c) y=ex ln x
d) y=ln (cos x )
e) y=√3 x2+ x+1
2) Differentiate the equation below.
a) y=5 x2+4 x3
b) y=(2x2−x )2
c) y=5 x3+x2
x4+2
d) y=7−3 x2
3−x
e) y=sin (2 x+5 )
f) y=cos (x3−2 x+2 )
3) Derive the equation below:
a) y=(2x2+x ) (3 x−1 )3
b) y=sin√2 x−1c) y=5 ln (x3+x2−15 )
d) y= e5 x (x2−1 )
e) y= x2+12 x2−1
4) Using the suitable method
differentiate the following variables
a) y=3 x2+4 x−8
b) y=2cos (3x2+4 )
c) y=2x3 (4 x−2 )4
d) y=tan2 (x2−2x+3 )
e) y= 4 x2−2
3 x+1
f) y= 2e2 x−5
sin (3−4 x3 )
5) Differentiate the equation below.
a) y=(3 x2+ 72 x)3
b) y=√4 x2+1c) y=e−3xsin 2 x
d) y=3 x2 ln 4 x
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e) y=3 x2−xx+1
6) Derive the equation below:
a) y=2x4−x−3+ x−1
b) y= (2x+6 )4
c) y=x (4−x )2
d) y= x2−5x2−1
e) y=sin34 x
7) Find dydx for the following equations
a) y=x3+2 x2−4 x+5
b) y= (x+1 ) ( x−3 )
c) y=e3− x2
+x3
d) y=3 x2+5 x
2 x−4
e) y= (2x−5 )3 ( x+1 )
f) y= x2+2√ x+1
8) Find dydx for the following equations
a) y=7 x3+4 x−2+3 x−4
b) y=(2+ x2 )3
c) y=√x3+ 2x2d) y=4 x2 tan x
e) y=3 x2+1
cos x
9) Using the right method, differentiate
the functions given.
a) p=q2(2q+ 3q )b) y=(3x2−5 )7
c) y= 2x√ x+1
d) y=x2 (x2−3 )4
e) y=4e4 x
10) Derive the equation below:
a) y=√2 x+ xb) y=4 x2+ x−2−2 x−1
c) y= x2
2 x−1
d) y=( x3 ) (2 x−2 )
e) y=ln (sin x )
f) y=3sin (2 x2+1 )
11) Find the for the parametric
functions given below in terms of t.
a)
b)
c)
12) Find the second derivatives for the
function
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a)
b)
c)
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