Learning Goal

Syllabus Requirement

Students need to learn how to differentiate ??x^{n}??, and related sums and differences with the below expressions being highlighted in particular (as ones that students may be expected to differentiate):

•    ??(2x+5)(x-1)??; and
•    ??\frac{x^{2}+5x-3}{3x^{\frac{1}{2}}}??.

Study Notes

The rules for differentiating are shown in the box below:

The rules for differentiating.

Rule 1: How to differentiate ??x^{n}??
If ??f(x) = x^{n}?? then ??f'(x) = nx^{n-1}??.

Rule 2: Differentiating a constant
If ??f(x) = a??, where ??a?? is a constant then ??f'(x) = 0??.

This is a special case of Rule 1 - any constants will differentiate to ??0??.

Rule 3: How to differentiate ??ax^{n}??
If ??f(x) = ax^{n}??, where ??a?? is a constant then ??f'(x) = anx^{n-1}??.

Notice that the constant does not affect the differentiation.

Rule 4: Differentiating sums:
If ??f(x) = g(x) + h(x)?? then ??f'(x) = g'(x) + h'(x)??.

This rule means that if you have a complicated expression to differentiate, you can do it in 'chunks', by differentiating each part in turn.

Worked Examples

We'll now run through some examples of each of these rules in turn.

Rule 1

Example 1

Question

Differentiate ??f(x) = x^{2}??.

Example 2

Question

Differentiate ??f(x) = \sqrt{x}??.

Example 3

Question

Differentiate ??f(x) = \frac{1}{4\sqrt{x}}??.

Rule 2

Example 4

Question

Differentiate ??f(x) = 30??.

Rule 3

Example 5

Question

Differentiate ??f(x) = \frac{5}{x^4}??.

Rule 4

Example 6

Question

Differentiate ??f(x) = x^{2} + 3x + 20??.

Example 7

Question

Differentiate ??y = \frac{x-1}{x^3}??.

Example 8

Question

Differentiate ??f(x) = (2x-3)(3x+\frac{1}{x})??.

Exam Questions

The table below contains every exam question that has been asked on this topic, this includes normal papers from both January and June sittings, International papers and Specimen papers.

To see an exam question and solution simply click on the load question icon (), the question will appear below the table and the solution can be shown by clicking the "Show Solution" button that also appears.

The full exam paper and mark schemes are also available for download by clicking on the download icons in each row of the table ().

Question
Exam Board Subject Paper Year Month Module Question
No.
Parts Total
Marks
Exam
Paper
Mark
Scheme
Edexcel Maths Standard 2005 January Core 1 2 (i)(a), (i)(b) 4
Edexcel Maths Standard 2005 June Core 1 2 (a) 2
Edexcel Maths Standard 2006 January Core 1 4 (a) 2
Edexcel Maths Standard 2006 June Core 1 5 (a), (b) 7
Edexcel Maths Standard 2007 January Core 1 1 n/a 4
Edexcel Maths Standard 2007 January Core 1 8 (a) 3
Edexcel Maths Standard 2007 June Core 1 3 (a), (b) 4
Edexcel Maths Standard 2008 January Core 1 5 (b) 4
Edexcel Maths Standard 2008 June Core 1 4 (a), (b) 5
Edexcel Maths Standard 2009 January Core 1 6 (b) 4
Edexcel Maths Standard 2009 June Core 1 3 (a) 3
Edexcel Maths Standard 2009 June Core 1 9 (b), (c) 5
Edexcel Maths Standard 2010 January Core 1 1 n/a 3
Edexcel Maths Standard 2010 January Core 1 6 (a) 4
Edexcel Maths Standard 2010 June Core 1 7 n/a 6
Edexcel Maths Standard 2011 January Core 1 11 (a) 4
Edexcel Maths Standard 2011 June Core 1 2 (a) 3
Edexcel Maths Standard 2011 June Core 1 10 (b) 3
Edexcel Maths Standard 2012 January Core 1 1 (a) 3
Edexcel Maths Standard 2012 January Core 1 8 (a) 2
Edexcel Maths Standard 2012 June Core 1 4 (a), (b) 6
Edexcel Maths Standard 2013 January Core 1 11 (a) 3
Edexcel Maths Standard 2013 June Core 1 9 (b) 2
Edexcel Maths International 2013 June Core 1 1 n/a 4

Exam Tips

1. Exam questions typically consist of applying the above rules to more complex expressions, for example, you may need to rewrite expressions, before differentiating, by:
Converting surds to index notation (e.g. ??\sqrt{x} = x^{\frac{1}{2}}??)
Simplifying fractional expressions (e.g. ??\frac{x^{2}}{x^{-1}}??)
Multiplying out brackets.

2. Sometimes exam questions will ask you to evaluate a differentiated expression at a certain value, e.g. ??f'(3)??.

3. Be particularly careful with negative indexes as it is very easy to make arithmetic errors.

4. Whether you use ??\frac{dy}{dx}?? or ??f'(x)?? depends on what is used in the question, but you should use the same notation.