Download Mathematics with calculus mac301 - Math-HS

MATHEMATICS WITH CALCULUS
MAC301
•Course
information
•Intro to Differentiation
•Review of year 12 skills
•Assignment 1.
DIFFERENTIATION BY SIGHT
Derivative/ derived
function/ gradient
function
Function
f(x)=axn
y=
Differentation
axn
f
’(x)=naxn-1
dy/dx=naxn-1
To
differentiate is to find the equation of the gradient function of any curve.
Gradients
The
describe the instantaneous rate of change
process of finding the derivative is called differentiation.
Functions
that can be differentiated are said to be differentiable.
DIFFERENTIATION REVIEW
1.
Powers of x
2.
5.
Coefficients
Constant terms
Multiple terms
Products
6.
Quotients
7.
Negative indices
Surds/ fractional indices
3.
4.
8.
y = x3
y=x
y = 4πx3
y=5
y = x10-7x+5
y=x2(2x7-6)
2x3  5x 2
y
x
y = 3/x 7
y  3 x2
NB: The convention is to give answers in the same form as the question.
Exercise 3.1C A1 (page 15)
Differentiate the following functions:
yx
3
2 y x 9
1 2
3 y  x  x 1
2
1
4
11
y  10x  2x  3x  4x  19
4
5
3
5
6
7
8
9
10

y  3x
3
y x
2
y  (x  2)(x  3x  5)
2
y  (3x  2)
4
5x  6x
y
2
x
3
2x  4x  9
y
1
2
x
4
1
Differentiate
11
11 ─ 1 =
x
dy
10
 11x
dx
yx
10
11
2
Differentiate
3─1= 2
3
x ─9
dy
2
 3x
dx
y x 9
3
3
1 2
Differentiate y  x  x  1
2
1 122
 x + 1x + 1
2
dy
 x 1
dx
4
Differentiate
y  10x  2x  3x  4x  19
4
5
3
4 ґ 10x4- 1 + 5 ґ 2x5- 1 - 3 ґ 3x3- 1 - 1 ґ 4x1- 1 - 0
dy
3
4
2
 40x  10x  9x  4
dx
5
Differentiate
- 4 ґ 3x
y  3x
- 4- 1
dy
5
 12 x
dx
4
6
Differentiate y 
3
Write in index form:
yx
1
x
3
1
3
1
-1
3
dy 1
 x
dx 3
2
3
x
7 Differentiate
y  (x  2)(x  3x  5)
2
Expand
y  x  3x  5x  2x  6x  10
3
2
Simplify
y  x  5x  x  10
dy
2
 3x  10x  1
dx
3
2
2
8
Differentiate y  (3x  2)
Expand
2
y = 9x - 6x - 6x + 4
Simplify
y  9x  12x  4
dy
 18x  12
dx
2
2
5x  6x
Differentiate y 
2
x
2
Divide through by x
4
9
(Multiply each term in the numerator by x
y  5x  6x
2
1
dy
2
 10 x  6 x
dx
- 2
)
2x  4x  9
3
10 Differentiate y 
Divide through by x
(Same as multiplying by
y  2x
5
dy
 5x
dx
3
2
2
 4x
 2x
1

x
2
1
2
1
x
1
2
2
- 1
2 )
 9x

1
9
 x
2
2

3
2