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Introduction to Differentiation (DP IB Maths: AI HL)
Revision Note
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
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 .
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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 the power of is multiplied by a constant then the derivative is also multiplied by that constant
- If then where and is a constant
- The alternative notation (to) is to use
- If then
- e.g. If then
- 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 roots will need to be rewritten as fractional powers of first
- e.g. If then rewrite as and differentiate
- 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 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 then
- 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
Examiner Tip
- A common mistake is not simplifying expressions before differentiating
- The derivative of can not be found by multiplying the derivatives of and
Worked example
The function is given by
, where
Find the derivative of
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