Rate of change /
Differentiation (2)
•Gradient
of curves
•Differentiating
Recall: A bit of new symbology
y
x
dy
dx
dy
dx
= “difference in y” = gradient of line
“difference in x”
PRONOUNCED “dee-why by dee-ex”
Gradient of Curves
y=x2
The tangent to
the curve gives
the gradient at
that point
y
Zoom
(3,9)
x
Gradient = “difference in y”
“difference in x”
= 9.61 - 9
3.1 - 3
= 6.1
(3.1,3.12) B
(3.1,9.61)
A (3,9)
Gradient of Curves
y=x2
The tangent to
the curve gives
the gradient at
that point
y
Zoom
(3,9)
x
Gradient = “difference in y”
“difference in x”
= 9.0601 - 9
3.01 - 3
= 6.01
(3.01,3.012) B
(3.01,9.0601)
A (3,9)
x
3
3.1
3.01
3.001
3.0001
3.00001
3.000001
3.0000001
3.00000001
3.000000001
y
9
9.61
9.0601
9.006001
9.00060001
9.00006
9.000006
9.0000006
9.00000006
9.000000006
dx
dy
0.1 0.61
0.01 0.06
0 0.01
0
0
0
0
0
0
0
0
0
0
0
0
dy/dx
6.1
6.01
6.001
6.0001
6.00001
6.000001
6.0000001
6
6
As the interval in x decreases
it tends to a definite value always twice ‘x’
A bit of theory
y
x (delta x) is the
difference in the x
coordinates
Gradient =
x
y
x
As x gets smaller, it gives the gradient of the tangent dy
dx
More Terminology
dy
dx
is the symbol used for the gradient of the curve
The process of finding dy is called differentiating
dx
dy
The gradient function dx is known as the
derivative
Graphs of displacement and gradient vs time
s
The curves of gradient are
always one
power lesss (in
s
x) than the original curves
t
t
“y=mx+c”
“y=ax2+bx+c”
ds
dt
ds
dt
t
“y=ax3+bx2+cx+d”
ds
dt
t
t
“y=const.”
“y=mx+c”
t
“y=ax2+bx+c”
Lets do
some differentiating
The general rule (very important) is :-
n
x
If y =
dy = nxn-1
“Times by the
power and
reduce the
power by 1”
dx
E.g. if y = x2
dy
= 2x
dx
E.g. if y = x3
dy
= 3x2
dx
E.g. if y = 5x4
dy
= 5 x 4x3
dx
dy
3
=
20x
dx
Example 1
E.g. if y = x3 + 13x
dy
= 3x2 +13
dx
You can just add
them together
So the gradient at x=3 is …..
dy
dx
= 3 x 32 +13 = 27 + 13 = 40
You just substitute the x value in
Find dy for these functions :-
Gradient at
x=2
dx
3x2
dy
= 6x
dx
= 12
y=
x6
dy
= 6x5
dx
= 192
y=
5x5
dy
= 25x4
dx
= 400
y=
12x10
dy
= 120x9
dx
=
y=
x3
y=
+
x2
y = 6x3 + 3x2 + 11x
dy = 3x2 + 2x
= 12 +4 = 16
dx
dy
2 + 6x + 11
=
18x
dx
=72+12+11 =95
Harder Examples
E.g. if y = 3x(2x2 +9)
Expand bracket first
y = 6x3 +27x
dy = 18x2 +27
dx
5x 4  8x 2
y
2x
5x 4 8x 2
y

2x 2x
5 3
Divide
y  x  4x
through
2
dy 15 2
 x 4
dx 2
1
y x
x
Express as
fractional or
negative indices
1
dy 1  2
 x  x 2
dx 2
1
2
yx x
1
The rules
still work
dy
1
1

 2
dx 2 x x
Harder Examples - your turn
E.g. if y = 2x2(3x3 +x)
Expand bracket first
y = 6x5 +2x3
dy = 30x4 +6x2
dx
10 x 3  x 2
y
4x
10 x 3 x 2
y

4x 4x
5 2 1 Divide
y x  x
2
4 through
dy
1
 5x 
dx
4
3
3
y x 2
x
Express as
fractional or
negative indices
2
dy 1  3
 x  6 x 3
dx 3
1
3
y  x  3x
2
The rules
still work
dy
1
6

 3
dx 33 x 2 x