Differentiation [Mathematics]

Note: This tutorial mainly targets O/L & A/L students.

What does differentiation mean?
I’d like to say, the purpose of differentiation is to find the derivative i.e. (dY/dX)
For your information: X and Y can be substituted with any other letter!
dY/dX is the gradient. And the gradient always tells you how Y changes with respect to X

• The basic rule for differentiating is, if y=x^n then dy/dx = nx^(n-1)

This is going to be more than just a tutorial. Let’s make this a quiz. I will do 10 questions along with detailed workings. First try it yourself then check the answer(s). Ok? Great!
Lets assume that ^ means to-the-power, / means division and |||| means to-seperate.

Questions

Differentiate:

1. Y = 3x^3 – (1/x)

2. Y = x + (1/2x)

3. Y = √x + (1/2√x)

4. Y = x^2(x+5)^2

5. Y = 10 – (5x/33) + (x^2/100)

6. Y = (3x + 6x^2 + x^4) / 3x^4

7. Y = (1/x + x)^2

8. Y = 2x(1/2x + 2)^2

9. Y = (x^2/3) – (3/x^2)

10. Y = (x^2 + x) / x

Answers

1. Y = 3x^3 – (1/x)
= 3x^3 – x^-1
dy/dx = 9x^2 + x^-2 OR 9x^2 + 1/x^2

2. Y = x + (1/2x)
= x + 1/2x^-1
dy/dx = 1 – 1/2x^-2 OR 1 – 1/2x^2

3. Y = √x + 1/2√x
= x^1/2 + 1/2x^1/2 = x^1/2 + 1/2x^-1/2
dy/dx = 1/2x^-1/2 – 1/4x^-3/2

4. Y = x^2(x+5)^2 |||| (x+5)^2 = x^2 + 10x + 25
= x^2(x^2 + 10x + 25) = x^4 + 10x^3 + 25x^2
dy/dx = 4x^3 + 30x^2 + 50x

5. Y = 10 – (5x/33) + (x^2/100)
dy/dx = -5/33 + 2x/100 |||| -5/33 + x/50

6. Y = (3x + 6x^2 + x^4) / 3x^4
= 1/x^3 + 2/x^2 + 1/3 {by cancellation}
= x^-3 + 2x^-2 + 1/3
dy/dx = -3x^-4 – 4x^-3 OR -3/x^4 – 4/x^3

7. Y = (1/x + x)^2
= x^2 + 1/x^2 + 2 = x^2 + x^-2 + 2
dy/dx = 2x – 2x^-3 OR 2x – 2/x^3

8. Y = 2x(1/2x + 2)^2
= 2x(1/4x^2 + 2/x + 4)
= 1/2x + 8x + 4 = 1/2x^-1 + 8x + 4
dy/dx = -1/2x^-2 + 8

9. Y = (x^2/3) – (3/x^2)
= x^2/3 – 3x^-2
dy/dx = 2x/3 + 6x^-3 OR 2x/3 + 6/x^3

10. Y = (x^2 + x) / x
= x + 1 {by cancellation}
dy/dx = 1

Do leave a comment if you find it hard to read; we’ll explain :]
– thanks for your comments

• AL
• Math
• mathematics
• OL