Functions and algebra: Investigate and use instantaneous rate of change of a variable when interpreting models both in mathematical and real life situations
Subject outcome
Subject outcome 2.4: Investigate and use instantaneous rate of change of a variable when interpreting models both in mathematical and real life situations
Learning outcomes
 Establish the derivatives of the following functions from first principles:
 [latex]\scriptsize f\left( x \right)=b;~f\left( x \right)=ax+b;f\left( x \right)=a{{x}^{2}}+b;f\left( x \right)={{x}^{3}};f\left( x \right)=a{{x}^{3}};f\left( x \right)=\displaystyle \frac{1}{x};f\left( x \right)=\displaystyle \frac{a}{x}[/latex]
Note: The binomial theorem does not form part of the curriculum.
 [latex]\scriptsize f\left( x \right)=b;~f\left( x \right)=ax+b;f\left( x \right)=a{{x}^{2}}+b;f\left( x \right)={{x}^{3}};f\left( x \right)=a{{x}^{3}};f\left( x \right)=\displaystyle \frac{1}{x};f\left( x \right)=\displaystyle \frac{a}{x}[/latex]
 Find the derivatives of the functions in the form:
 [latex]\scriptsize f\left( x \right)=a{{x}^{n}}[/latex]
 [latex]\scriptsize f\left( x \right)=a\ln kx[/latex]
 [latex]\scriptsize f\left( x \right)=a{{e}^{{kx}}}[/latex]
 [latex]\scriptsize f\left( x \right)=\text{asin}kx[/latex]
 [latex]\scriptsize f\left( x \right)=\text{acos}kx[/latex]
 [latex]\scriptsize f\left( x \right)=\text{atan}kx[/latex]
Where:

 [latex]\scriptsize f\left( x \right)=a{{x}^{n}}~~~~~~~~~~~{f}'\left( x \right)=na{{x}^{{n1}}}~[/latex]
 [latex]\scriptsize f\left( x \right)=a\ln kx~~~~{f}'\left( x \right)=\displaystyle \frac{k}{x}[/latex]
 [latex]\scriptsize f\left( x \right)=a{{e}^{{kx}}}~~~~~~~~{f}'\left( x \right)=k{{e}^{{kx}}}[/latex]
 [latex]\scriptsize f\left( x \right)=\text{asin}kx~~~{f}'\left( x \right)=ka\cos kx~~[/latex]
 [latex]\scriptsize f\left( x \right)=\text{acos}kx~~~~{f}'\left( x \right)=ka\sin kx~~[/latex]
Examples to include are:

 [latex]\scriptsize 3{{x}^{2}};\displaystyle \frac{3}{{{{x}^{{3}}}}};~\displaystyle \frac{2}{{\sqrt[3]{{{{x}^{2}}}}}};2\ln 3x;\displaystyle \frac{1}{2}{{e}^{{2x}}};2\text{sin}3x;\displaystyle \frac{1}{3}\cos \displaystyle \frac{x}{2};~4\text{tan}x;\text{etc}.[/latex]
 Use the constant, sum and/or difference, product, quotient and chain rules for differentiation. Note: Combinations of rules in the same problem are excluded.
 Find the equation of the tangent to a graph at a specific point.
 Solve practical problems involving rates of change. Note: velocity and acceleration may be included.
 Draw graphs of cubic functions by determining:
 yintercept
 roots (xintercepts)
 turning points using derivatives
 Determine/prove maximum and minimum turning points by making use of second order derivatives (Only: quadratic and cubic functions)
 Determine the point of inflection of cubic graphs by using second order derivatives.
Unit 1 outcomes
By the end of this unit you will be able to:
 Define the derivative using limits.
 Calculate the derivative from first principles.
Unit 2 outcomes
By the end of this unit you will be able to:
 Use various forms of notation to represent the derivative.
 Apply the power rule.
 Find the derivative of a constant.
 Find the derivative of a constant multiplied by a function.
 Find the derivative of a sum/difference.
 Find the derivative of a product.
 Find the derivative of a quotient.
 Apply the chain rule.
Unit 3 outcomes
By the end of this unit you will be able to:
 Define a tangent line.
 Determine the gradient of the tangent at a point.
 Find the equation of a tangent.
Unit 4 outcomes
By the end of this unit you will be able to:
 Solve practical problems involving rates of change.
Unit 5 outcomes
By the end of this unit you will be able to:
 Determine the shape of a cubic function.
 Determine the x and y intercepts.
 Find the turning points of the graph.
 Find the maximum and minimum values of the graph.
 Find the point of inflection and discuss concavity using second derivatives.