6A - Review of differentiation
In VCE Mathematical Methods Unit 2, we looked at two types of rates of change:
- The average rate of change was found by determining the gradient between two points on a curve.
- The instantaneous rate of change was found by determining the gradient of a tangent line to a curve.
■ The derivative is a function that can determine the gradient of a tangent line at any point on a curve. The derivative function is found by the process of differentiation.
In order to find the instantaneous rate of change, we require a process that will allow us to find the gradient of a tangent at any point, \(P\), of a function, \(f\). The process of finding the derivative function is differentiation.
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■ For a function, \(f\), the derivative function can be found using differeniation from first principles \[f'(x)=lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}\]
6A - Example 1: Calculating the derivative from first principles (From Unit 2)
Differentiate the function \(f(x)=3x\) using first principles.
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6A - Example 1: Video solution
6A - Example 1: Practice
Question 1: ABC Question 2: ABC 6A - Example 1: Solutions
Question 1: ABC Question 2: ABC |
6A - Example 2: Calculating the derivative from first principles (From Unit 2)
Differentiate the function \(f(x)=-2x^2+6x\) using first principles.
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6A - Example 2: Video solution
6A - Example 2: Practice
Question 1: ABC Question 2: ABC 6A - Example 2: Solutions
Question 1: ABC Question 2: ABC |
6A - Example 3: Calculating the derivative from first principles (From Unit 2)
Differentiate the function \(f(x)=\frac{1}{x}\) using first principles.
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6A - Example 3: Video solution
6A - Example 3: Practice
Question 1: ABC Question 2: ABC 6A - Example 3: Solutions
Question 1: ABC Question 2: ABC |
6A - Example 4: Graphs and their derivatives (VCAA, 2006)
[VCAA, 2006 Exam 2 MC Question 14]
For the graph of \(y=f(x)\) shown above, \(f'(x)\) is negative when
A. \(-3<x<3\) B. \(-3\leq x \leq 3\) C. \(x<-3\) or \(x>3\) D. \(x\leq -3\) or \(x\geq 3\) E. \(-5<x<1\) or \(x>4\) |
6A - Example 4: Video solution
6A - Example 4: Practice
Question 1: ABC 6A - Example 4: Solutions
Question 1: ABC |
6A - Example 5: Graphs and their derivatives (VCAA, 2007)
[VCAA, 2007 Exam 2 MC Question 13]
For the graph of \(y=4x^3+27x^2-30x+10\) the subset of \(R\) for which the gradient is negative is given by the interval A. \((0,5,5.0)\) B. \((-4.99,0.51)\) C. \((-\infty, \frac{1}{2})\) D. \((-5,\frac{1}{2})\) E. \((2.25,\infty)\) |
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.
Differentiation from First Principles - Worksheet A
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Past VCAA examination questions have, typically, not examined differentiation from first principles.
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.