1. Rate of change / Differentiation (2)
Gradient of curves
Differentiating

2. dy dx Recall: A bit of new symbology y x dy dx = “difference in y”
“difference in x”
= gradient of line
PRONOUNCED “dee-why by dee-ex”

3. Gradient of Curves
y=x2
The tangent to the curve gives the gradient at that point y
Zoom
(3,9) x
(3.1,3.12) B
(3.1,9.61)
Gradient = “difference in y”
“difference in x” =
= 6.1
A (3,9)

4. Gradient of Curves
y=x2
The tangent to the curve gives the gradient at that point y
Zoom
(3,9) x
(3.01,3.012) B
(3.01,9.0601)
Gradient = “difference in y”
“difference in x” =
= 6.01
A (3,9)

5. As the interval in x decreases it tends to a definite value - always twice ‘x’

6. A bit of theory
x (delta x) is the difference in the x coordinates y y
Gradient = x x dy dx
As x gets smaller, it gives the gradient of the tangent

7. More Terminology dy dx
is the symbol used for the gradient of the curve
The process of finding is called differentiating dy dx dy dx
The gradient function is known as the derivative

8. Graphs of displacement and gradient vs time
The curves of gradient are always one power less (in x) than the original curves s t s t s t
“y=ax2+bx+c”
“y=ax3+bx2+cx+d”
“y=mx+c” ds dt ds dt ds dt t t t
“y=const.”
“y=mx+c”
“y=ax2+bx+c”

9. If y = xn = nxn-1 Lets do some differentiating dy dx
The general rule (very important) is :-
If y = xn
“Times by the power and reduce the power by 1” dy dx
= nxn-1
E.g. if y = x2
= 2x dy dx
E.g. if y = x3
= 3x2 dy dx
E.g. if y = 5x4
= 5 x 4x3
= 20x3 dy dx

10. You just substitute the x value in
Example 1
E.g. if y = x3 + 13x
= 3x2 +13 dy dx
You can just add them together
So the gradient at x=3 is ….. dy dx =
3 x = = 40
You just substitute the x value in

11. Find for these functions :- Gradient at x=2dy dx
Find for these functions :-
Gradient at
x=2 dy dx
y= 3x2
y = x6
y = 5x5
y = 12x10
y = x3 + x2
y = 6x3 + 3x2 + 11x
= 6x
= 6x5
= 25x4
= 120x9
= 3x2 + 2x
= 18x2 + 6x + 11
= 12
= 192
= 400 =
= = 16
= =95

12. Harder Examples y = 3x(2x2 +9) Expand bracket first
E.g. if y = 3x(2x2 +9)
y = 6x3 +27x
= 18x2 +27 dy dx
Divide through
Express as fractional or negative indices
The rules still work

13. Harder Examples - your turn
y = 2x2(3x3 +x)
Expand bracket first
E.g. if
y = 6x5 +2x3
= 30x4 +6x2 dy dx
Divide through
Express as fractional or negative indices
The rules still work