Differentiation – SS2 Further Mathematics Lesson Note (Third Term)

Differentiation is the process of finding the derivative, dy/dx, of a function y = f(x). The derivative is the limit of Δy/Δx as Δx approaches 0. It can be found from first principles or by the general formula: if y = axⁿ, then dy/dx = anxⁿ⁻¹.

Behavioural objectives

By the end of the lesson, students should be able to:

• state the meaning of the limit of a function, including the right hand and left hand limits;
• find the limit of a function where the numerator and denominator both approach 0;
• explain what it means to differentiate from first principles;
• use the general formula to differentiate functions of the form y = axⁿ and sums of such terms.

What are limits of a function?

Notation for the limiting value

lim_x→a f(x) = L

Intuitively, this means that as x gets closer to a, f(x) gets closer and closer to L.

Right hand limit

lim_x→a⁺ f(x) = L

Intuitively, this means that as x gets closer to a from the right, f(x) gets closer and closer to L.

Left hand limit

lim_x→a⁻ f(x) = L

Intuitively, this means that as x gets closer to a from the left, f(x) gets closer and closer to L.

Worked example: finding a limit

Example: Find lim_x→5 (x² − 25)/(x − 5).

Solution

The numerator and denominator both approach 0. However, x² − 25 = (x + 5)(x − 5).

Hence, (x² − 25)/(x − 5) = x + 5.

Thus, lim_x→5 (x² − 25)/(x − 5) = lim_x→5 (x + 5)(x − 5)/(x − 5) = lim_x→5 (x + 5) = 5 + 5 = 10.

How do you differentiate from first principles?

Consider the function:

y = f(x) … (1)

Let Δy be a small increase in y due to a small increase Δx in x. Then:

y + Δy = f(x + Δx) … (2)

Subtract (1) from (2):

y + Δy − y = f(x + Δx) − f(x)

Δy = f(x + Δx) − f(x)

Divide both sides by Δx:

Δy/Δx = [f(x + Δx) − f(x)]/Δx

Take the limit as Δx → 0:

lim_Δx→0 Δy/Δx = dy/dx

Note: To differentiate by first principles means to work strictly from the ordinary definition or the derived definition, without using any other theorem.

How do you differentiate using the general formula?

For any function y = axⁿ, where a and n are constants, the derivative of y with respect to x is:

dy/dx = anxⁿ⁻¹

dy/dx is read as “dee y by dee x”.

Example 1

Differentiate y = x⁶ with respect to x.

Solution

y = x⁶

dy/dx = 6x⁵

Example 2

Differentiate y = 4x⁴ − 3x³ − 2x² + x − 22 with respect to x.

Solution

If y = 4x⁴ − 3x³ − 2x² + x − 22, then

dy/dx = 4 × 4x⁴⁻¹ − 3 × 3x³⁻¹ − 2 × 2x²⁻¹ + 1 × 1x¹⁻¹ − 22 × 0x⁰⁻¹

dy/dx = 16x³ − 9x² − 4x + 1

Evaluation

1. Explain what it means to differentiate from first principles.
2. Find lim_x→3 (x² − 9)/(x − 3).
3. Differentiate y = x⁸ with respect to x.
4. Differentiate y = 5x³ − 2x² + 7x − 9 with respect to x.

Assignment

1. Find lim_x→2 (x² − 4)/(x − 2).
2. Differentiate y = 3x⁵ + 2x³ − x² + 6x − 10 with respect to x.