1. Differentiation from First Principles
Learning Objective: to understand that differentiation is the process for calculating the gradient of a curve. The rate of change can be calculated from first principles by considering the limit of the function at any one point.

2. Rates of change
This graph shows the distance that a car travels over a period of 5 seconds.
The gradient of the graph tells us the rate at which the distance changes with respect to time.
time (s)
distance (m) 5 40
In other words, the gradient tells us the speed of the car.
The car in this example is travelling at a constant speed since the gradient is the same at every point on the graph.

3. Rates of change
In most situations, however, the speed will not be constant and the distance–time graph will be curved.
For example, this graph shows the distance–time graph as the car moves off from rest.
time (s)
distance (m)
The speed of the car, and therefore the gradient, changes as you move along the curve.
To find the rate of change in speed we need to find the gradient of the curve.
The process of finding the rate at which one variable changes with respect to another is called differentiation.
In most situations this involves finding the gradient of a curve.

4. The gradient of a curve
The gradient of a curve at a point is given by the gradient of the tangent at that point.
Look at how the gradient changes as we move along a curve:
Explain that the direction of a curve changes as we move along it. Demonstrate this by moving the point along the curve. Observe where the gradient of the tangent at the point is positive, negative and 0.

5. Differentiation from first principles
Suppose we want to find the gradient of a curve at a point A.
We can add another point B on the line close to point A.
δx represents a small change in x and δy represents a small change in y.
As point B moves closer to point A, the gradient of the chord AB gets closer to the gradient of the tangent at A.
Explain that the small letter delta, δ, is used to represent a small change in the variable. A capital delta, Δ, can also be used.

6. Differentiation from first principles
We can write the gradient of the chord AB as:
As B gets closer to A, δx gets closer to 0 and gets closer
to the value of the gradient of the tangent at A.
δx can’t actually be equal to 0 because we would then have division by 0 and the gradient would then be undefined.
It is important that students are clear that delta is used to represent a small change in the variable. It is not itself a variable and certainly cannot be cancelled out. You may need to explain the meaning of infinitesimal, perhaps by referring to the infinite.
Explain that at the limit, the gradient of the chord will actually be equal to the gradient of the tangent. As the chord approaches the gradient, δx approaches 0, so that we can treat it as zero. The idea of a limit can be a difficult one to grasp. It is, however, central to the understanding of the principles of differential calculus. Differentiation from first principles is not examined in this module and so it is enough for students to understand the basic concept.
Instead we must consider the limit as δx tends to 0.
This means that δx becomes infinitesimal without actually becoming 0.

7. Differentiation from first principles
If A is the point (3, 9) on the curve y = x2 and B is another point close to (3, 9) on the curve, we can write the coordinates of B as (3 + δx, (3 + δx)2).
The gradient of chord AB is:
B(3 + δx, (3 + δx)2) δy
When differentiating from first principles, δx is sometimes replaced by the variable h. dy/dx is then given by the limit as h tends to 0.
A(3, 9) δx

8. Differentiation from first principles
At the limit where δx → 0, 6 + δx → 6.
We write this as: 6
So the gradient of the tangent to the curve y = x2 at the point (3, 9) is 6.
Let’s apply this method to a general point on the curve y = x2.
If we let the x-coordinate of a general point A on the curve
y = x2 be x, then the y-coordinate will by x2.
So, A is the point (x, x2).
If B is another point close to A(x, x2) on the curve, we can write the coordinates of B as (x + δx, (x + δx)2).

9. Differentiation from first principles
The gradient of chord AB is:
B(x + δx, (x + δx)2) δy
A(x, x2) δx
This result tells us that the gradient of any point on the curve y = x2 is 2 times the x-coordinate.

10. The gradient function
So the gradient of the tangent to the curve y = x2 at the general point (x, y) is 2x.
2x is often called the gradient function or the derived function of y = x2.
If the curve is written using function notation as y = f(x), then the derived function can be written as f ′(x).
So, if:
f(x) = x2
f ′(x) = 2x
Then:
This notation is useful if we want to find the gradient of f(x) at a particular point.
For example, the gradient of f(x) = x2 at the point (5, 25) is:
f ′(5) = 2 × 5 = 10

11. Now we shall differentiate y = x3 from first principles:

12. Using the notation We have shown that for y = x3dy dx
We have shown that for y = x3
is usually written as
then:
So if y = x3
represents the derivative of y with respect to x.
You may wish to point out that many other notations are employed to represent derivatives. However, the notations f ′(x) and dy/dy are the only ones used at this level.
Stress that dy/dy is the rate of change in y with respect to x.
of the tangent.
Remember, is the gradient of a chord, while is the gradient

13. Using the notation dy dx
This notation can be adapted for other variables so, for example:
represents the derivative of s with respect to t.
If s is distance and t is time then we can interpret this as the rate of change in distance with respect to time. In other words, the speed.
Also, if we want to differentiate 2x4 with respect to x, for example, we can write:
We could work this out by differentiating from first principles, but in practice this is unusual.

14. Task 1
Differentiate from first principles
y = x4
y = 1/x