# Ch 12: Derivatives of Functions

### About This Chapter

## Derivatives of Functions - Chapter Summary

The short, professionally designed lessons in this chapter explain the derivatives of functions. You'll review continuity as it applies to a function, define limits using a graph, and examine the processes used to determine the limits of functions. Other topics include the slopes of secant and tangent lines and differentiation from first principles. This chapter is designed to help you:

- Calculate derivatives by applying the rules of differentiation
- Differentiate complex functions using the chain rule
- Calculate derivatives of constant functions
- Give the formal definition of a derivative
- Follow the correct process to calculate the derivatives of polynomial equations

Even the most challenging topics are easy to understand and follow with our fun learning materials. Our video and text lessons are all accompanied by multiple-choice quizzes you can use to ensure you've understood what you've learned. If you need additional help, you're welcome to reach out to one of our instructors. We've made it easy to study whenever the activity works best for you with our mobile-friendly tools, accessible 24 hours a day.

### 1. Continuity in a Function

Travel to space and explore the difference between continuous and discontinuous functions in this lesson. Learn how determining continuity is as easy as tracing a line.

### 2. Using a Graph to Define Limits

My mom always said I tested the limits of her patience. Use graphs to learn about limits in math. You won't get grounded as we approach limits in this lesson.

### 3. How to Determine the Limits of Functions

You know the definition of a limit. You know the properties of limits. You can connect limits and continuity. Now use this knowledge to calculate the limits of complex functions in this lesson.

### 4. Slopes of Tangent & Secant Lines

Being able to find the slope of both tangent lines and secant lines allows us to calculate the rate of change of a curve. In this lesson, we learn how to do both, as well as learning that secant lines can only provide an average whereas tangent lines can be exact.

### 5. Derivatives: The Formal Definition

The derivative defines calculus. In this lesson, learn how the derivative is related to the instantaneous rate of change with Super C, the cannonball man.

### 6. Applying the Rules of Differentiation to Calculate Derivatives

In this lesson, we'll review common derivatives and their rules, including the product, quotient and chain rules. We'll also examine how to solve derivative problems through several examples.

### 7. Using the Chain Rule to Differentiate Complex Functions

If you've ever seen a complicated function, this lesson is for you. Most functions that we want to differentiate are complicated functions, for which no single derivative rule will work. In this lesson, learn how to use the chain rule to simplify nesting equations.

### 8. Calculating Derivatives of Constant Functions

In calculus, one of the primary tasks is to find the derivative of a function. Different types of functions require different techniques to find their derivative. This lesson will cover the simple process to find derivatives of constant functions.

### 9. Calculating Derivatives of Polynomial Equations

Polynomials can describe just about anything and are especially common in describing motion. Learn the tricks to quickly finding the derivatives of these ubiquitous functions.

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### Other Chapters

Other chapters within the HSC Mathematics: Exam Prep & Syllabus course

- Mathematical Proofs & Reasoning
- Basic Arithmetic & Algebra
- Factorization of Binomials & Trinomials
- Types of Equations & Inequalities
- Plane Geometry
- Circle Plane Geometry
- Probability & Randomization
- Arithmetic & Geometric Series
- Trigonometric Ratios
- Geometrical Representations of Functions
- Linear Functions & Lines
- Quadratic Polynomials & Parabolas
- Geometrical Applications of Differentiation
- Integration
- Logarithmic & Exponential Functions
- Calculating Trigonometric Functions
- Understanding Inverse Functions
- Factoring & Graphing Polynomials
- Roots & Coefficients of Polynomials
- Binomial Theorem & Probability
- Graphing & Solving Functions
- Solving & Graphing Complex Numbers
- Geometric Representations of Complex Numbers
- Square Roots, Powers & Roots of Complex Numbers
- Conic Sections Basics
- The Ellipse in Algebra
- The Hyperbola
- The Rectangular Hyperbola
- Calculus Applications: Rate of Change
- Calculus Applications: Velocity & Acceleration
- Calculus Applications: Projectile & Harmonic Motion
- Calculus Applications: Resisted Motion
- Calculus Applications: Circular Motion
- HSC Mathematics Flashcards