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Introduction to Differentiation (Cambridge O Level Additional Maths)
Revision Note
Definition of Gradient
What is the gradient of a curve?
- At a given point the gradient of a curve is defined as the gradient of the tangent to the curve at that point
- A tangent to a curve is a line that just touches the curve at one point but doesn't cut the curve at that point
- A tangent may cut the curve somewhere else on the curve
- It is only possible to draw one tangent to a curve at any given point
- Note that unlike the gradient of a straight line, the gradient of a curve is constantly changing
Examiner Tip
- If a question asks for the "rate of change of ..." then it is asking for the "gradient"
Worked example
The diagram shows the curve with equation The tangent, , to the curve at the point is also shown.
Using the diagram, calculate the gradient of the curve at .
The gradient of the curve at the point A is the same as the gradient of the tangent T.
Calculate the gradient of the line.
The gradient is 3
Definition of Derivatives
What is a derivative?
- Calculus is about rates of change
- the way a car’s position on a road changes is its speed (velocity)
- the way the car’s speed changes is its acceleration
- The gradient (rate of change) of a (non-linear) function varies with
- The derivative of a function is a function that relates the gradient to the value of
- For example, the derivative of is
- This means that when , the gradient of is
- And when , the gradient of is
- For example, the derivative of is
- The derivative is also called the gradient function
Worked example
The derivative of is .
Use the derivative to find the gradient of at the point .
Substitute into the derivative,
gradient = 3
Note that the answer is the same as in the method above
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Differentiating Powers of x
What is differentiation?
- Differentiation is the process of finding an expression for 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
Examiner 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 factorise 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
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