IB DP Maths: AA HL复习笔记5.2.3 Higher Order Derivatives
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
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
- a second order derivative is the rate of change of the rate of change of a function
- 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
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
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
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