Introduction to Derivatives
- Before introducing a derivative, an understanding of a limit is helpful
What is a limit?
- The limit of a function is the value a function (of
) approaches as
approaches a particular value from either side
- Limits are of interest when the function is undefined at a particular value
- For example, the function
will approach a limit as
approaches 1 from both below and above but is undefined at
as this would involve dividing by zero
What might I be asked about limits?
- You may be asked to predict or estimate limits from a table of function values or from the graph of
- You may be asked to use your GDC to plot the graph and use values from it to estimate a limit
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
- The derivative is also called the gradient function
How are limits and derivatives linked?
- Consider the point
on the graph of
as shown below
is a series of chords
- The gradient of the function
at the point
is equal to the gradient of the tangent at point
- The gradient of the tangent at point
is the limit of the gradient of the chords
as point
‘slides’ down the curve and gets ever closer to point
- The gradient of the function changes as
changes
- The derivative is the function that calculates the gradient from the value
What is the notation for derivatives?
- For the function
, the derivative, with respect to
, would be written as
- Different variables may be used
- e.g. If
then
- e.g. If
What might I be asked about derivatives?
- You may be asked to use the graphing features of your GDC to find the gradients of a function at different values of
- From a series of gradient values, you may be asked to suggest an expression for the derivative (gradient function) of a function
Worked Example
The graph of where
passes through the points
and
.
a)
Find the gradient of the chords
and
.
b)
Estimate the gradient of the tangent to the curve at the point
.
c)
Use your GDC to find the gradient of the tangent at the pont
.
Differentiating Powers of x
What is differentiation?
- Differentiation is the process of finding an expression of the derivative (gradient function) from the expression of a function
How do I differentiate powers of x?
- Powers of
are differentiated according to the following formula:
- If
then
where
- This is given in the formula booklet
- If
- If the power of
is multiplied by a constant then the derivative is also multiplied by that constant
- If
then
where
and
is a constant
- If
- The alternative notation (to
) is to use
- If
then
- e.g. If
then
- e.g. If
- If
- Don't forget these two special cases:
- If
then
- e.g. If
then
- e.g. If
- If
then
- e.g. If
then
- e.g. If
- These allow you to differentiate linear terms in
and constants
- If
- Funcitons involving roots will need to be rewritten as fractional powers of
first
- e.g. If
then rewrite as
and differentiate
- e.g. If
- 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
- e.g. If
How do I differentiate sums and differences of powers of x?
- The formulae for differentiating powers of
apply to all rational powers so it is possible to differentiate any expression that is a sum or difference of powers of
- e.g. If
then
- e.g. If
- Products and quotients cannot be differentiated in this way so would need expanding/simplifying first
- e.g. If
then expand to
which is a sum/difference of powers of
and can be differentiated
- e.g. If
What might I be asked to do once I’ve found the derivative (gradient function)?
- 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
- A GDC can be used to help find gradients or verify answers
- Take extra care when differentiating negative and fractional powers of
but
Exam Tip
- A common mistake is not simplifying expressions before differentiating
- The derivative of
can not be found by multiplying the derivatives of
and
- The derivative of
Worked Example
The function is given by
where .
a)
Find the derivative of
.
![5-1-1-ib-sl-ai-aa-extra-we2-soltn-a](../../../../../../../../cdn-cgi/image/index-323.htm?f=auto,width=1920/uploads/2022/03/5-1-1-ib-sl-ai-aa-extra-we2-soltn-a.png)
b)
Find the gradient of the tangent to the curve
at the point where
.