Differentiating Powers of x
What is differentiation?
- Differentiation is the process of finding an expression of the derivative (gradient function) from the equation of a curve
- The equation of the curve is written and the gradient function is written
How do I differentiate powers of x?
- Powers of are differentiated according to the following formula:
- If then
- e.g. If then
- you "bring down the power" then "subtract one from the power"
- If then
- Don't forget these two special cases:
- If then
- e.g. If then
- If then
- e.g. If then
- These allow you to differentiate linear terms in and constants
- If then
- Functions involving fractions with denominators in terms of will need to be rewritten as negative powers of first
- e.g. If then rewrite as and differentiate
How do I differentiate sums and differences of powers of x?
- The formulae for differentiating powers of work for a sum or difference of powers of
- e.g. If then
- This is sometimes referred to differentiating 'term-by-term'
- e.g. If then
- Products and quotients (divisions) cannot be differentiated in this way so they need expanding/simplifying first
- e.g. If then expand to which is a sum/difference of powers of and can then be differentiated
What can I do with derivatives (gradient functions)?
- The derivative can be used to find the gradient of a function at any point
- The gradient of a function at a point is equal to the gradient of the tangent to the curve at that point
- A question may refer to the gradient of the tangent
Exam Tip
- Don't try to do too many steps in your head; write the expression in a format that you can differentiate before you actually differentiate it
- e.g. can be rewritten as which is then far easier to differentiate
Worked example
Find the derivative of
Rewrite the term
Apply the rule for differentiating powers () and apply the special cases for the terms and 8 ( and )
Unless a question specifies there is not usually a need to rewrite/simplify the answer
This is a product of two (equal) brackets so cannot be differentiated directly
Expand the brackets to get an expression in powers of
Take time to get the expansion correct, writing stages out in full if necessary
Differentiate 'term-by-term', looking out for those special cases
There is a factor of 4 but there is no demand to factoise the final answer in the question
This is a quotient so cannot be differentiated directly
Spot the single denominator which means we can split the fraction by the two terms on the numerator
Simplify using the laws of indices
Each term is now a power of , so differentiate 'term-by-term'
There is demand to simplify or write the answer in a particular form