Introduction to Differentiation (DP IB Maths: AA HL)

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Paul

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8 June 2023

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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 $x$ ) approaches as $x$ approaches a particular value from either side
- Limits are of interest when the function is undefined at a particular value
- For example, the function $f (x) = \frac{x^{4} - 1}{x - 1}$ will approach a limit as $x$ approaches 1 from both below and above but is undefined at $x = 1$ 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 $y = f (x)$
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 $x$
The derivative of a function is a function that relates the gradient to the value of $x$
The derivative is also called the gradient function

How are limits and derivatives linked?

Consider the point $P$ on the graph of $y = f (x)$ as shown below
- $[P Q_{i}]$ is a series of chords

5-1-2-definiton-of-derivatives-diagram-1

The gradient of the function $f (x)$ at the point $P$ is equal to the gradient of the tangent at point $P$
The gradient of the tangent at point $P$ is the limit of the gradient of the chords $[P Q_{i}]$ as point $Q$ ‘slides’ down the curve and gets ever closer to point $P$
The gradient of the function changes as $x$ changes
The derivative is the function that calculates the gradient from the value $x$

What is the notation for derivatives?

For the function $y = f (x)$ , the derivative, with respect to $x$ , would be written as

$\frac{d y}{d x} = f' (x)$

Different variables may be used
- e.g. If $V = f (s)$ then $\frac{d V}{d s} = f' (s)$

Worked example

The graph of $y = f (x)$ where $f (x) = x^{3} - 2$ passes through the points $P (2, 6), A (2.3, 10.167), B (2.1, 7.261)$ and $C (2.05, 6.615125)$ .

Find the gradient of the chords

[P A], [P B]

and

[P C]

5-1-1-ib-sl-ai-aa-we1-soltn-a

Estimate the gradient of the tangent to the curve at the point

P

5-1-1-ib-sl-ai-aa-we1-soltn-b

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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 $x$ are differentiated according to the following formula:
- If $f (x) = x^{n}$ then $f' (x) = n x^{n - 1}$ where $n \in ℚ$
- This is given in the formula booklet
If the power of $x$ is multiplied by a constant then the derivative is also multiplied by that constant
- If $f (x) = a x^{n}$ then $f' (x) = a n x^{n - 1}$ where $n \in ℚ$ and $a$ is a constant
The alternative notation (to $f' (x)$ ) is to use $\frac{d y}{d x}$
- If $y = a x^{n}$ then $\frac{d y}{d x} = a n x^{n - 1}$
  - e.g. If $y = - 4 x^{\frac{1}{2}}$ then $\frac{d y}{d x} = - 4 \times \frac{1}{2} \times x^{\frac{1}{2} - 1} = - 2 x^{- \frac{1}{2}}$
Don't forget these two special cases:
- If $f (x) = a x$ then $f' (x) = a$
  - e.g. If $y = 6 x$ then $\frac{d y}{d x} = 6$
- If $f (x) = a$ then $f' (x) = 0$
  - e.g. If $y = 5$ then $\frac{d y}{d x} = 0$
- These allow you to differentiate linear terms in $x$ and constants
Functions involving roots will need to be rewritten as fractional powers of $x$ first
- e.g. If $f (x) = 2 \sqrt{x}$ then rewrite as $f (x) = 2 x^{\frac{1}{2}}$ and differentiate
Functions involving fractions with denominators in terms of $x$ will need to be rewritten as negative powers of $x$ first
- e.g. If $f (x) = \frac{4}{x}$ then rewrite as $f (x) = 4 x^{- 1}$ and differentiate

How do I differentiate sums and differences of powers of x?

The formulae for differentiating powers of $x$ apply to all rational powers so it is possible to differentiate any expression that is a sum or difference of powers of $x$
- e.g. If $f (x) = 5 x^{4} - 3 x^{\frac{2}{3}} + 4$ then
  $f' (x) = 5 \times 4 x^{4 - 1} - 3 \times \frac{2}{3} x^{\frac{2}{3} - 1} + 0$
  $f' (x) = 20 x^{3} - 2 x^{- \frac{1}{3}}$
Products and quotients cannot be differentiated in this way so would need expanding/simplifying first
- e.g. If $f (x) = (2 x - 3) (x^{2} - 4)$ then expand to $f (x) = 2 x^{3} - 3 x^{2} - 8 x + 12$ which is a sum/difference of powers of $x$ and can be differentiated

Examiner Tip

A common mistake is not simplifying expressions before differentiating
- The derivative of $(x^{2} + 3) (x^{3} - 2 x + 1)$ can not be found by multiplying the derivatives of $(x^{2} + 3)$ and $(x^{3} - 2 x + 1)$

Worked example

The function $f (x)$ is given by

$f (x) = 2 x^{3} + \frac{4}{\sqrt{x}}$ , where $x > 0$

Find the derivative of

f (x)

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