1. difference in the y-values
e.g. Find the gradient of the line joining the points with coordinates and
Solution:
difference in the y-values x
7 - 1 = 6 x
3 - 1 = 2
difference in the x-values

2. This branch of Mathematics is called Calculus
The gradient of a straight line is given by
We use this idea to get the gradient at a point on a curve
Gradients are important as they measure the rate of change of one variable with another. For the graphs in this section, the gradient measures how y changes with x
This branch of Mathematics is called Calculus

3. The Gradient at a point on a Curve
Definition: The gradient of a point on a curve equals the gradient of the tangent at that point.
e.g.
Tangent at
(2, 4) x 12 3
The gradient of the tangent at (2, 4) is
So, the gradient of the curve at (2, 4) is 4

4. The gradient changes as we move along a curve

7. The Rule for Differentiation

8. The Rule for Differentiation

11. Differentiation from first principles
f(x+h) A
f(x) x
x + h
Gradient of AP =

12. Differentiation from first principles
f(x+h) A
f(x) x
x + h
Gradient of AP =

13. Differentiation from first principles
f(x+h) A
f(x) x
x + h
Gradient of AP =

14. Differentiation from first principles
f(x+h) A
f(x) x
x + h
Gradient of AP =

15. Differentiation from first principles
f(x+h) A
f(x) x
x + h
Gradient of AP =

16. Differentiation from first principles
f(x) x
Gradient of tangent at A =

17. Differentiation from first principles
f(x) = x2

18. Differentiation from first principles
f(x) = x3

19. Generally with
h is written as δx
And f(x+ δx)-f(x) is written as δy

20. Generally

21. We need to be able to find these points using algebra
Points with a Given Gradient
e.g. Find the coordinates of the points on the curve
where the gradient equals 4
Gradient of curve
= gradient of tangent
= 4
We need to be able to find these points using algebra

24. Exercises
Find the coordinates of the points on the curves with the gradients given 1.
where the gradient is -2
Ans: (-3, -6) 2.
where the gradient is 3
( Watch out for the common factor in the quadratic equation )
Ans: (-2, 2) and (4, -88)

25. Increasing and Decreasing Functions
An increasing function is one whose gradient is always greater than or equal to zero.
for all values of x
A decreasing function has a gradient that is always negative or zero.
for all values of x

26. e.g.1 Show that is an increasing function
Solution:
is the sum of
a positive number ( 3 )  a perfect square ( which is positive or zero for all values of x, and
a positive number ( 4 )
for all values of x
so, is an increasing function

27. e.g.2 Show that is an increasing function.
Solution:
for all values of x

28. The graphs of the increasing functions and are

29. Exercises
1. Show that is a decreasing function and sketch its graph.
2. Show that is an increasing function and sketch its graph.
Solutions are on the next 2 slides.

30. Solutions
1. Show that is a decreasing function and sketch its graph.
Solution: This is the product of a square which is always
and a negative number,
so for all x. Hence is a decreasing function.

31. Solutions
2. Show that is an increasing function
Solution:
Completing the square:
which is the sum of a square which is
and a positive number. Hence y is an increasing function.

32. (-1, 3) on line: The equation of a tangent
e.g. 1 Find the equation of the tangent at the point
(-1, 3) on the curve with equation
Solution:
The gradient of a curve at a point and the gradient of the tangent at that point are equal
(-1, 3) x
Gradient = -5
At x = -1
(-1, 3) on line:
So, the equation of the tangent is