1. Differentiation

2. Introduction You will learn what Differentiation is
You will see how to apply it to solve graph based problems
It is one of the single most important topics there is in Maths!

3. Teachings for Exercise 7B

4. You need to be able to find the gradient function of a formula
Differentiation
You need to be able to find the gradient function of a formula
On a straight line graph, the gradient is constant, the same everywhere along the line.
On any curved graph, the gradient is always changing. Its value depends on where you are along the x-axis.
The different gradients can be shown by tangents. These are lines that touch the curve in only one place.
Differentiation is the process whereby we can find a formula to give the gradient on a curve, at any point on it.
Straight Line = Constant Gradient
tangent
Curved line = Gradient Changes
tangent 7B

5. You need to be able to find the gradient function of a formula
Differentiation
You need to be able to find the gradient function of a formula
As a general rule, if;
then…
Examples
Find the derived function of each of the following… a)
f(x) = x3
f’(x) = 3x2
This is the formula for the curve, ie) the function b)
f(x) = 2x2
f’(x) = 4x1 (4x) c)
f(x) = x-2
This is the gradient function for f(x). The formula that gives the gradient at any point.
Its is also know as the derivative, or ‘derived function’
f’(x) = -2x-3 d)
f(x) = -3x-3
f’(x) = 9x-4 7B

6. You need to be able to find the gradient function of a formula
Differentiation
You need to be able to find the gradient function of a formula
As a general rule, if;
then…
Examples
Find the derived function of each of the following… e)
Must be written in the form axn first!
This is the formula for the curve, ie) the function
This is the gradient function for f(x). The formula that gives the gradient at any point.
Its is also know as the derivative, or ‘derived function’ f)
Must be written in the form axn first! 7B

7. Teachings for Exercise 7C

8. Differentiation
You need to be able to solve Graphical problems using the Gradient Function
Remember that differentiating gives us a formula for the gradient at a given point on the graph (x).
A standard question will ask you to work out the gradient of a curve at a particular point. This is when differentiating is used.
Examples
a) Calculate the gradient of the curve f(x) = x2 where x = 3
Differentiate to get the gradient function
Substitute in the value for x at the required point
b) Calculate the gradient of the curve f(x) = x2 at the coordinate (-2,4)
y = x2
Differentiate to get the gradient function
Substitute in the value for x at the required point 7C

9. Differentiation
You need to be able to solve Graphical problems using the Gradient Function
Remember that differentiating gives us a formula for the gradient at a given point on the graph (x).
A standard question will ask you to work out the gradient of a curve at a particular point. This is when differentiating is used.
Examples
c) Find dy/dx when y equals x2 – 6x - 4
Differentiate each term separately.
A number on its own disappears
This also means the gradient function 7C

10. Differentiation
You need to be able to solve Graphical problems using the Gradient Function
Remember that differentiating gives us a formula for the gradient at a given point on the graph (x).
A standard question will ask you to work out the gradient of a curve at a particular point. This is when differentiating is used.
Examples
d) Find dy/dx when y equals 3 – 5x2.
Differentiate each term separately.
A number on its own disappears 7C

11. Differentiate to get the gradient function
Differentiation
You need to be able to solve Graphical problems using the Gradient Function
Remember that differentiating gives us a formula for the gradient at a given point on the graph (x).
A standard question will ask you to work out the gradient of a curve at a particular point. This is when differentiating is used.
Examples
e) Let f(x) = 4x2 – 8x + 3. Find the gradient of the curve y = f(x) at the point (1/2, 0)
Differentiate to get the gradient function
Substitute in x = 1/2 7C

12. Differentiation
You need to be able to solve Graphical problems using the Gradient Function
Remember that differentiating gives us a formula for the gradient at a given point on the graph (x).
A standard question will ask you to work out the gradient of a curve at a particular point. This is when differentiating is used.
Examples
f) Find the coordinates when the graph y = 2x2 - 5x + 3 has a gradient of 7.
Differentiate to get the gradient function
The gradient is 7 at the point we want
Add 5
Divide by 4
The x coordinate where the gradient is 7 has a value of 3. Substitute this into the ORIGINAL function to find the y-coordinate
So the graph has a gradient of 7 at (3,6) 7C

13. Differentiation
You need to be able to solve Graphical problems using the Gradient Function
Remember that differentiating gives us a formula for the gradient at a given point on the graph (x).
A standard question will ask you to work out the gradient of a curve at a particular point. This is when differentiating is used.
Examples
g) Find the gradient of the curve y = x2 where it meets the line y = 4x - 3
Set the equations equal to each other (to represent where they meet)
Group on the left
Factorise
The lines will meet at (3,9) and (1,1) by substitution.
Differentiate y = x2 to get the gradient function
At (3,9) the gradient will be 6 (by putting ‘3’ into the gradient function)
At (1,1) the gradient will be 2 (by putting ‘1’ into the gradient function) 7C

14. Teachings for Exercise 7E

15. Differentiation
You need to be able to deal with much more complicated equations when differentiating
Remember as before, all terms must be written in the form axn before they can be differentiated.
It is useful to note that at this stage, being able to rewrite in this way is not essential. However being able to switch between forms will be very useful on harder questions.
Being able to do this now is worth practising as you will definitely need it on C2/3/4!
Examples
Differentiate the following: a)
Rewrite in the form axn
Differentiate
Factorise 7E

16. Differentiation
You need to be able to deal with much more complicated equations when differentiating
Remember as before, all terms must be written in the form axn before they can be differentiated.
It is useful to note that at this stage, being able to rewrite in this way is not essential. However being able to switch between forms will be very useful on harder questions.
Being able to do this now is worth practising as you will definitely need it on C2/3/4!
Examples
Differentiate the following: b)
Rewrite in the form axn
Differentiate
Imagine the term was split apart
Rewrite the x term using power rules
Group the fractions by multiplying tops/bottoms 7E

17. Differentiation
You need to be able to deal with much more complicated equations when differentiating
Remember as before, all terms must be written in the form axn before they can be differentiated.
It is useful to note that at this stage, being able to rewrite in this way is not essential. However being able to switch between forms will be very useful on harder questions.
Being able to do this now is worth practising as you will definitely need it on C2/3/4!
Examples
Differentiate the following: c)
Split into 2 fractions
Cancel x’s on the first one
Rewrite in the form axn
Differentiate
Rewrite using power rules
Make the Denominators common 7E

18. Teachings for Exercise 7F

19. Differentiation or or 7F
You can repeat the process of differentiation to get the ‘second order derivative’
Examples
Original Equation
Differentiate once (first order derivative) or
Rewrite in the form axn
Differentiate
Differentiate again
Differentiate twice (second order derivative) or 7F

20. Differentiation or or 7F
You can repeat the process of differentiation to get the ‘second order derivative’
Examples
Original Equation
Differentiate once (first order derivative) or
Rewrite in the form axn
Differentiate
Differentiate twice (second order derivative) or
Differentiate again 7F

21. Teachings for Exercise 7H

22. Differentiation
You can use differentiation to find the tangent to a curve at a particular point, as well as the normal at that point
Remember the curve we have is based on an equation
The tangent is a straight line that intersects the curve at on point only.
The gradient of the tangent is the same as the gradient of the curve at the point given (so you can differentiate to get it)
The ‘normal’ is a straight line perpendicular to the tangent where is touches the curve.
Normal
Curve
Tangent 7H

23. Differentiation
You can use differentiation to find the tangent to a curve at a particular point, as well as the normal at that point
Find the equation of the tangent to the curve y = x3 – 3x2 + 2x - 1, at the point (3,5).
So we need the gradient of the tangent. It will be the same as the gradient of the curve at the point given (x = 3)
Differentiate to get the gradient function
Substitute the x value in to find the gradient
The gradient at (3,5) is 11
Use the formula for a straight line!
Substitute the co-ordinate and gradient in
Expand bracket
Curve
Add 5
Tangent 7H

24. Differentiation
You can use differentiation to find the tangent to a curve at a particular point, as well as the normal at that point
Find the equation of the normal to the curve y = 8 - 3√x at the point where x = 4.
Start by finding the gradient of the curve at that point.
Gradient = -3/4 (this is of the tangent)
The Normal is perpendicular to the tangent
Gradient of the Normal = 4/3
Rewrite
Differentiate
Rewrite for substitution
Sub in x = 4
The Gradient where x = 4 is -3/4 7H

25. Differentiation
You can use differentiation to find the tangent to a curve at a particular point, as well as the normal at that point
Find the equation of the normal to the curve y = 8 - 3√x at the point where x = 4.
Gradient of the Normal = 4/3
Now we need the co-ordinates at that point. We already have x = 4.
Co-ordinates at the intersection = (4,2)
Substitute in x = 4
3 times 2 = 6!
Use the formula for a straight line!
Substitute the co-ordinate and gradient in
Multiply by 3
Expand bracket
Rearrange 7H

26. Showing how differentiation works (sort of!)
y = x2
Showing how differentiation works (sort of!)
Multiply the bracket
This is a lowercase ‘delta’, representing a small increase in x
Gradient = change in y ÷ change in x
Group some terms
Cancel δx
(x+δx, (x+δx)2)
(x+δx)2 – x2
At the original point, δx = 0
x+δx - x
(x, x2)

27. Showing how differentiation works (sort of!)
y = f(x)
Showing how differentiation works (sort of!)
Simplify the denominator
This is a lowercase ‘delta’, representing a small increase in x
Gradient = change in y ÷ change in x
(x+δx, f(x+δx))
f(x+δx) – f(x)
x+δx - x
This is the ‘definition’ for differentiating a function (you won’t really need this specifically until C3)
(x, f(x))

28. Summary We have learnt what Differentiation is, and the process used
We have also seen the appropriate notation
We have also seen when to use it