# Functions and algebra: Investigate and use instantaneous rate of change

### Subject outcome

Subject outcome 2.5: Investigate and use instantaneous rate of change

### Learning outcomes

• Determine limits of functions intuitively.
• Distinguish between the value of a function at a particular point and the limit of that function at that point.
• Establish the derivatives of the following functions from first principles:
$\scriptsize f\left( x \right)=b,~~f\left( x \right)=x,~~f\left( x \right)=ax+b,~~f\left( x \right)={{x}^{2}},~~f\left( x \right)=a{{x}^{2}}+b$
• Find the derivative of functions in the form:
• $\scriptsize \displaystyle f(x)=a{{x}^{n}}\text{or }y=a{{x}^{n}},{f}'(x)=n.a{{x}^{{(n-1)}}}\text{or }\displaystyle \frac{{dy}}{{dx}}=n.a{{x}^{{(n-1)}}}$ Examples to include: $\scriptsize 4{{x}^{2}};~\displaystyle \frac{3}{{{{x}^{{-3}}}}};\displaystyle \frac{2}{{\sqrt[3]{{{{x}^{2}}}}}};\displaystyle \frac{5}{{3{{x}^{2}}}}$ (all examples within this range).
• Use the constant, sum and/or difference rule by first simplifying the expression.
• If $\scriptsize y=f\left( x \right)=a~$ and $\scriptsize a$is a constant function then: $\scriptsize \displaystyle \frac{{dy}}{{dx}}={f}'\left( x \right)=0~$
• If $\scriptsize y=\text{k}f\left( x \right)\text{ }\!\!~\!\!\text{ }~$then $\scriptsize \displaystyle \frac{d}{{dx}}\left[ {\text{k}f\left( x \right)} \right]=\text{k}\displaystyle \frac{d}{{dx}}\left[ {f\left( x \right)} \right]$
• If $\scriptsize y=f\left( x \right)\pm g\left( x \right)~~$then $\scriptsize \displaystyle \frac{d}{{dx}}\left[ {f\left( x \right)\pm g\left( x \right)} \right]=\displaystyle \frac{d}{{dx}}\left[ {f\left( x \right)\left] {\pm \displaystyle \frac{d}{{dx}}} \right[g\left( x \right)} \right]$.
• Solve maxima and minima problems about real life situations from given equations.

### Unit 1 outcomes

By the end of this unit you will be able to:

• Determine limits of functions intuitively.
• Distinguish between the value of a function at a particular point and the limit of the function at that point.

### Unit 2 outcomes

By the end of this unit you will be able to:

• Develop an intuitive understanding of the gradient at a point.
• Differentiate using first principles.

### Unit 3 outcomes

By the end of this unit you will be able to:

• Apply the constant, sum and difference rules for differentiation.
• Use notation appropriately.

### Unit 4 outcomes

By the end of this unit you will be able to:

• Use differentiation to find optimal solutions.