# 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:
• $\scriptsize f\left( x \right)=b;~f\left( x \right)=ax+b;f\left( x \right)=a{{x}^{2}}+b;f\left( x \right)={{x}^{3}};f\left( x \right)=a{{x}^{3}};f\left( x \right)=\displaystyle \frac{1}{x};f\left( x \right)=\displaystyle \frac{a}{x}$
Note: The binomial theorem does not form part of the curriculum.
• Find the derivatives of the functions in the form:
• $\scriptsize f\left( x \right)=a{{x}^{n}}$
• $\scriptsize f\left( x \right)=a\ln kx$
• $\scriptsize f\left( x \right)=a{{e}^{{kx}}}$
• $\scriptsize f\left( x \right)=\text{asin}kx$
• $\scriptsize f\left( x \right)=\text{acos}kx$
• $\scriptsize f\left( x \right)=\text{atan}kx$

Where:

• $\scriptsize f\left( x \right)=a{{x}^{n}}~~~~~~~~~~~{f}'\left( x \right)=na{{x}^{{n-1}}}~$
• $\scriptsize f\left( x \right)=a\ln kx~~~~{f}'\left( x \right)=\displaystyle \frac{k}{x}$
• $\scriptsize f\left( x \right)=a{{e}^{{kx}}}~~~~~~~~{f}'\left( x \right)=k{{e}^{{kx}}}$
• $\scriptsize f\left( x \right)=\text{asin}kx~~~{f}'\left( x \right)=ka\cos kx~~$
• $\scriptsize f\left( x \right)=\text{acos}kx~~~~{f}'\left( x \right)=-ka\sin kx~~$

Examples to include are:

• $\scriptsize 3{{x}^{2}};\displaystyle \frac{3}{{{{x}^{{-3}}}}};~-\displaystyle \frac{2}{{\sqrt{{{{x}^{2}}}}}};2\ln 3x;\displaystyle \frac{1}{2}{{e}^{{-2x}}};2\text{sin}3x;\displaystyle \frac{1}{3}\cos \displaystyle \frac{x}{2};~-4\text{tan}x;\text{etc}.$
• 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:
• y-intercept
• roots (x-intercepts)
• 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. 