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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