IB DP Maths: AI HL复习笔记5.2.4 Second 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
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
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- test for local minimum and maximum points
- help determine the nature of stationary points
- help determine the concavity of a function
- graph derivatives
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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
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