6D - Graphs of derivative functions
Calculating the derivative of a function results in another function which can also be graphed. In this section we will examine several important features of a graph of a function and the graph of its derivative function. Consider a function \(f(x)\) and its derivative function \(f'(x)\):
Graph of \(f(x)\) |
Graph of \(f'(x)\) |
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The graphs of \(f(x)=\frac{1}{3}x^3-4x\) and its derivative \(f'(x)=x^2-4\) are shown to the right. Consider the key features of \(f(x)\) and the associated features on \(f'(x)\):
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6D - Example 1: Sketching the graph of a derivative function (VCAA, 2007)
[VCAA, 2007 Exam 1 Question 3]
The diagram below shows the graph of a function with domain \(R\).
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6D - Example 1: Video solution
6D - Example 1: Practice
Question 1: ABC 6D - Example 1: Solutions
Question 1: ABC |
6D - Example 2: Graphs and their derivatives (VCAA, 2009)
[VCAA, 2009 Exam 2 MC Question 8]
For the function \(f:R\rightarrow R, f(x)=(x+5)^2(x-1)\), the subset of \(R\) for which the gradient of \(f\) is negative is A. \((-\infty,1)\) B. \((-5,1)\) C. \((-5,-1)\) D. \((-\infty,-5)\) E. \((-5,0)\) |
6A - Example 2: Video solution
6A - Example 2: Practice
Question 1: ABC 6A - Example 2: Solutions
Question 1: ABC |
6D - Example 3: Graphs and their derivatives (VCAA, 2009)
[VCAA, 2009 Exam 2 MC Question 21]
A cubic function has rule \(y=f(x)\). The graph of the derivative function \(f'\) crosses the \(x\)-axis at \((2,0)\) and \((-3,0)\). The maximum value of the derivative function is \(10\). The value of \(x\) for which the graph of \(y=f(x)\) has a local maximum is A. \(-2\) B. \(2\) C. \(-3\) D. \(3\) E. \(\frac{1}{2}\) |
6A - Example 3: Video solution
6A - Example 3: Practice
Question 1: ABC 6A - Example 3: Solutions
Question 1: ABC |
6D - Example 4: Graphs and their derivatives (VCAA, 2010)
[VCAA, 2010 Exam 2 MC Question 16]
The gradient of the function \[f:R\rightarrow R, f(x)=\frac{5x}{x^2+3}\] is negative for A. \(-\sqrt{3}<x<\sqrt{3}\) B. \(x>3\) C. \(x\in R\) D. \(x<-\sqrt{3}\) and \(x>\sqrt{3}\) E. \(x<0\) |
6A - Example 4: Video solution
6A - Example 4: Practice
Question 1: ABC 6A - Example 4: Solutions
Question 1: ABC |
6D - Example 5: Graphs and their derivatives (VCAA, 2010)
[VCAA, 2010 Exam 2 MC Question 19]
The graph of the gradient function \(y=f'(x)\) is shown below. Which of the following could represent the graph of the function \(f(x)\)?
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6A - Example 5: Video solution
6A - Example 5: Practice
Question 1: ABC 6A - Example 5: Solutions
Question 1: ABC |
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Additional Exercises
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Topic Worksheets
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VCAA Questions
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Other Resources
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Please view Topic Worksheets for any additional questions available at this point in time.
Graphs of derivative functions - Worksheet A (Coming Soon)
Coming soon
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Past VCAA examination questions relating to graphs of derivative functions will be collated here in the future.
Other resources from external sources may be collated here in the future.
If you know of any good resources that could be included here, please send them via email.
If you know of any good resources that could be included here, please send them via email.