Differentiation of Algebraic Function – SS3 Mathematics Lesson Note (Second Term)

Differentiation is the process of finding the derivative of a function. The derivative is also the gradient function of the curve. If y = f(x), the derivative dy/dx = f′(x) is the limiting value of the change in y divided by the change in x, as the change in x tends to zero.

Behavioural objectives

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

• state the meaning of differentiation and the notations used for a derivative;
• find the derivative of a simple function from the first principle;
• state the rule d/dx (xⁿ) = nxⁿ⁻¹;
• use the rule to differentiate algebraic functions such as 5x² – 3x and 2x³ – 3x + 8.

What is differentiation?

This lesson covers:

1. Differentiation of algebraic functions: meaning of differentiation
2. Differentiation from the first principle
3. Standard derivatives of some basic functions

Consider the curve whose equation is y = f(x). Recall that the gradient of the chord AB is m = (y₂ – y₁)/(x₂ – x₁) = (f(x + δx) – f(x))/δx.

As point B moves close to A, δx becomes smaller and tends to zero. The limiting value is written as lim (δx → 0) of (f(x + δx) – f(x))/δx, and it is denoted by f′(x).

f′(x) is called the derivative of f(x) and the gradient function of the curve. The process of finding the derivative of f(x) is called differentiation.

Notations for the derivative

The notations commonly used for the derivative of a function are:

• f′(x), read as “f prime of x”;
• df/dx, read as “dee-f, dee-x”;
• dy/dx, read as “dee-y, dee-x”.

If y = f(x), then dy/dx = f′(x). It is called the differential coefficient of y with respect to x.

What is differentiation from the first principle?

The process of finding the derivative of a function from the consideration of the limiting value is called differentiation from the first principle.

Example 1

Find from the first principle the derivative of y = x².

Solution

1. y = x²
2. y + δy = (x + δx)² = x² + 2xδx + (δx)²
3. δy = x² + 2xδx + (δx)² – y = x² + 2xδx + (δx)² – x² = 2xδx + (δx)²
4. δy = (2x + δx)δx, so δy/δx = 2x + δx
5. As δx → 0, dy/dx = 2x

Example 2

Find from the first principle the derivative of 1/x.

Solution

1. Let y = 1/x
2. y + δy = 1/(x + δx)
3. δy = 1/(x + δx) – 1/x = (x – (x + δx))/((x + δx)x) = (x – x – δx)/(x² + xδx) = -δx/(x² + xδx)
4. δy/δx = -1/(x² + xδx)
5. As δx → 0, dy/dx = -1/x²

What are the rules of differentiation?

Let y = xⁿ. Then y + δy = (x + δx)ⁿ.

y + δy = xⁿ + nxⁿ⁻¹δx + [n(n – 1)/2!] xⁿ⁻²(δx)² + … + (δx)ⁿ

δy = nxⁿ⁻¹δx + [n(n – 1)/2!] xⁿ⁻²(δx)² + … + (δx)ⁿ

δy/δx = nxⁿ⁻¹ + [n(n – 1)/2!] xⁿ⁻²δx + …

As δx → 0, dy/dx = nxⁿ⁻¹.

Hence dy/dx = nxⁿ⁻¹ if y = xⁿ.

Example 3

Find the derivative of the following with respect to x: (a) x⁷ (b) x^½ (c) 5x² – 3x (d) -3x² (e) y = 2x³ – 3x + 8

Solution

1. Let y = x⁷. dy/dx = 7x⁷⁻¹ = 7x⁶
2. Let y = x^½. dy/dx = ½x^{½ – 1} = ½x^-½ = 1/(2√x)
3. Let y = 5x² – 3x. dy/dx = 10x – 3
4. Let y = -3x². dy/dx = 2 × (-3)x²⁻¹ = -6x
5. Let y = 2x³ – 3x + 8. dy/dx = 3 × 2x³⁻¹ – 3 + 0 = 6x² – 3

Evaluation

After differentiation from the first principle

Find from the first principle the derivatives of y with respect to x:

1. y = 3x³
2. y = 7x²
3. y = 3x² – 5x

After the rules of differentiation

1. If y = 5x⁴, find dy/dx.
2. Given that y = 4x⁻¹, find y′.

General evaluation

1. Find, from first principles, the derivative of 4x² – 2 with respect to x.
2. Find the derivative of the following: (a) 3x³ – 7x² – 9x + 4 (b) 2x³ (c) 3/x
3. Using the idea of the difference between two squares, simplify 243x² – 48y².
4. Expand (2x – 5)(3x – 4).
5. If the gradient of y = 2x² – 5 is -12, find the value of y.

Assignment

Reading assignment

NGM for SS 3, Chapter 10, pages 82 to 88.

Weekend assignment: objective

1. Find the derivative of 5x³. (a) 10x² (b) 15x² (c) 10x (d) 15x³
2. Find dy/dx if y = 1/x³. (a) -3/x⁴ (b) 3/x⁴ (c) 4/x³ (d) -4/x³
3. Find f′(x) if f(x) = x³. (a) 3x (b) 3x² (c) ½x³ (d) 2x³
4. Find the derivative of 1/x. (a) 1/x² (b) -1/x² (c) -x (d) -x²
5. If y = -(2/3)x³, find dy/dx. (a) (4/3)x² (b) 2x² (c) -2x² (d) -2x

Weekend assignment: theory

1. Find from the first principle the derivative of y = x + 1/x.
2. Find the derivative of 2x² – 2/x³.