Differentiating Basic Functions (Edexcel IGCSE Further Pure Maths)
Revision Note
Written by: Roger B
Reviewed by: Dan Finlay
Differentiating Powers of x
What is differentiation?
Differentiation is the process of finding the derivative (gradient function) of a function
How do I differentiate powers of x?
Powers of are differentiated according to the following formula:
If then
Bring the power down in front as a multiplier
Then subtract 1 from the power
This formula is not on the exam formula sheet, so you need to remember it
If the power of term is multiplied by a constant
then the derivative is also multiplied by that constant
If then
The alternative notation (to) is to use
If then
Don't forget these two special cases:
If then
e.g. If then
If then
e.g. If (or if equals any constant) then
Functions involving roots will need to be rewritten as fractional powers
e.g.
rewrite as
then differentiate
Functions involving fractions with in the denominator will need to be rewritten as negative powers
e.g.
rewrite as
then differentiate
How do I differentiate sums and differences of powers of x?
The formulae can be used to differentiate sums or differences of powers of
Just differentiate term by term
e.g.
Products and quotients cannot be differentiated in this way
These need to be expanded/simplifying first
e.g.
Expand to
Then differentiate term by term
You can't just multiply the derivatives of and together!
These can also be differentiated using the product rule or quotient rule
Examiner Tips and Tricks
Be careful with negative and fractional powers
It's easy to make a mistake when subtracting 1 from these
Worked Example
The function is given by
, where
Find the derivative of .
Start by rewriting the term as a power of
By laws of indices,
Now differentiate as powers of
Differentiating Trig Functions
How do I differentiate sin and cos?
The derivative of is
The derivative of is is
If is multiplied by a constant then
the derivative of is
the derivative of is
These can be derived by using the chain rule
but it's easier (and quicker) just to remember them
None of these formulae are on the exam formula sheet, so you need to remember them
For calculus with trigonometric functions angles must be measured in radians
Make sure you know how to change the angle mode on your calculator
Examiner Tips and Tricks
As soon as you see a question involving differentiation and trigonometry
put your calculator into radians mode
Worked Example
(a) Given the function , find .
Use with
(b) A curve has the equation .
Find the gradient of the curve at the point where , giving your answer as an exact value.
Start by finding
Use with
Substitute into to find the gradient
Differentiating e^x
How do I differentiate exponentials?
The derivative of is
Note that is its own derivative!
If is multiplied by a constant then
the derivative of is
This can be derived by using the chain rule
but it's easier (and quicker) just to remember it
Examiner Tips and Tricks
Remember this is not a 'powers of ' differentiation
the derivative of is , NOT
Worked Example
A curve has the equation.
Find the gradient of the curve at the point where, giving your answer correct to 3 significant figures.
Differentiate using
Substitute into to find the gradient
Note that is the exact value answer
Use a calculator to find the decimal version
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