Higher Order Derivatives (DP IB Maths: AA HL)

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Second Order Derivatives

What is the second order derivative of a function?

If you differentiate the derivative of a function (i.e. differentiate the function a second time) you get the second order derivative of the function
There are two forms of notation for the second order derivative
- $y = f (x)$
- $\frac{d y}{d x} = f' (x)$ (First order derivative)
- $\frac{d^{2} y}{d x^{2}} = f'' (x)$ (Second order derivative)
Note the position of the superscript 2’s
- differentiating twice (so $d^{2}$ ) with respect to $x$ twice (so $x^{2}$ )
The second order derivative can be referred to simply as the second derivative
- Similarly, the first order derivative can be just the first derivative
A first order derivative is the rate of change of a function
- a second order derivative is the rate of change of the rate of change of a function
  - i.e. the rate of change of the function’s gradient
Second order derivatives can be used to
- test for local minimum and maximum points
- help determine the nature of stationary points
- help determine the concavity of a function
- graph derivatives

How do I find a second order derivative of a function?

By differentiating twice!
This may involve
- rewriting fractions, roots, etc as negative and/or fractional powers
- differentiating trigonometric functions, exponentials and logarithms
- using chain rule
- using product or quotient rule

Exam Tip

Negative and/or fractional powers can cause problems when finding second derivatives so work carefully through each term

Worked example

Given that $f (x) = 4 - \sqrt{x} + \frac{3}{\sqrt{x}}$

Find

f' (x)

and

f'' (x)

5-2-3-ib-sl-aa-only-second-order-we-soltn-a

Evaluate

f'' (3)

.
Give your answer in the form

a \sqrt{b}

, where

b

is an integer and

a

is a rational number.

5-2-3-ib-sl-aa-only-second-order-we-soltn-b

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Higher Order Derivatives

What is meant by higher order derivatives of a function?

Many functions can be differentiated numerous times
- The third, fourth, fifth, etc derivatives of a function are generally called higher order derivatives
It may not be possible, or practical to (algebraically) differentiate complicated functions more than once or twice
Polynomials will, eventually, have higher order derivatives of zero
- Since powers of x reduce by 1 each time

What is the notation for higher order derivatives?

The notation for higher order derivatives follows the logic from the first and second derivatives

$f^{(n)} (x)$ or $\frac{d^{n} y}{d x^{n}}$

except the ‘dash’ (prime) notation is replaced with numbers as this would become cumbersome after the first few

- e.g. the fifth derivative would be

$f^{(5)} (x)$ or $\frac{d^{5} y}{d x^{5}}$

How do I find a higher order derivative of a function?

By differentiating as many times as required!
This may involve
- rewriting fractions, roots, etc as negative and/or fractional powers
- differentiating trigonometric functions, exponentials and logarithms
- using chain rule
- using product or quotient rule

Exam Tip

If you are required to evaluate a higher order derivative at a specific point your GDC can help
- Typically a GDC will only work out the first and second derivative directly from the original function
- But, if you wanted the fourth derivative, say, you only need differentiate twice algebraically, then call this the ‘original’ function on your GDC

Worked example

It is given that $f (x) = \sin 2 x$ .

Show that

f^{4} (x) = 16 f (x)

Without further working, write down an expression for

f^{8} (x)

5-2-3-ib-hl-aa-only-we2b-soltn

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