Optional callout text right side.

**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*

Comments are closed.

good information, you write it very clean. I am very lucky to get this tips from you.

WordPress supports latex (perhaps with a plugin). It’ll help a lot in readability IMO.

BTW, Cool stuff on tutebox. Kudos!

Thanks 🙂

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