CALCULUS · TEXTBOOK LESSON
Calculus studies change and accumulation. Differentiation gives instantaneous rates of change and gradients; integration gives accumulated change and can recover functions from their rates of change.
Learning objectives
By the end of this lesson, you should be able to:
- Use standard derivative and integral notation.
- Differentiate or integrate common algebraic functions.
- Interpret derivatives as gradients or rates of change.
- Apply calculus to stationary points, areas or accumulated quantities.
1. Key vocabulary and definition
Definition
If y=f(x), the derivative f′(x)=dy/dx gives the instantaneous rate of change of y with respect to x. An indefinite integral ∫f(x)dx represents a family of antiderivatives.
derivativegradientrate of changeintegralconstant of integrationstationary point
2. Essential facts and rules
These are the facts you should know before attempting the worked example.
Key mathematical facts
- d/dx(x^n)=nx^(n-1).
- ∫x^n dx = x^(n+1)/(n+1)+C for n≠-1.
- At a differentiable stationary point, dy/dx=0.
3. Standard method
A reliable method helps prevent errors and makes your reasoning easy to follow.
1
Rewrite the function in a form suitable for standard rules.
2
Apply the relevant differentiation or integration rule term by term.
3
Simplify the resulting expression.
4
For applications, impose given conditions or solve equations such as f′(x)=0.
5
Interpret the result in context and check units where relevant.
4. Worked example
Worked example
Differentiate y = 3x³ - 5x² + 4x - 7.
1
Differentiate each term using d/dx(x^n)=nx^(n-1).
2
d/dx(3x³)=9x² and d/dx(-5x²)=-10x.
3
d/dx(4x)=4 and d/dx(-7)=0.
4
Therefore dy/dx = 9x² - 10x + 4.
5. Common mistakes
Watch out for these
- Starting calculations before deciding what the question is asking.
- Skipping important steps or changing notation midway through a solution.
- Accepting an answer without checking whether its size, sign or unit is sensible.
6. Video lesson
This topic is ready for a matching video lesson. The written textbook lesson is fully available now; additional videos can be added to the same lesson template.
7. Practice exercise
Complete these without looking at the answers first.
- Differentiate x⁵.
- Differentiate 4x³ - 2x + 9.
- Integrate 6x².
- Integrate 3x² + 4x with respect to x.
- For y=x²-6x+5, find the x-coordinate of the stationary point.
Show answers
- 5x⁴
- 12x² - 2
- 2x³ + C
- x³ + 2x² + C
- 3
8. Chapter summary
- Understand the underlying idea before memorising a procedure.
- Use precise mathematical notation and show a logical method.
- Always check the final result.