# 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:

[latex]\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[/latex] - Find the derivative of functions in the form:
- [latex]\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)}}}[/latex] Examples to include: [latex]\scriptsize 4{{x}^{2}};~\displaystyle \frac{3}{{{{x}^{{-3}}}}};\displaystyle \frac{2}{{\sqrt[3]{{{{x}^{2}}}}}};\displaystyle \frac{5}{{3{{x}^{2}}}}[/latex] (all examples within this range).

- Use the constant, sum and/or difference rule by first simplifying the expression.
- If [latex]\scriptsize y=f\left( x \right)=a~[/latex] and [latex]\scriptsize a[/latex]is a constant function then: [latex]\scriptsize \displaystyle \frac{{dy}}{{dx}}={f}'\left( x \right)=0~[/latex]
- If [latex]\scriptsize y=\text{k}f\left( x \right)\text{ }\!\!~\!\!\text{ }~[/latex]then [latex]\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][/latex]
- If [latex]\scriptsize y=f\left( x \right)\pm g\left( x \right)~~[/latex]then [latex]\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][/latex].

- 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.