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Definition of Differentiation

Sarah Miller

Sarah Miller

6 min read

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Study Guide Overview

This study guide covers the concept of derivatives and their relationship to tangents. It explains how to find the derivative of a function, how it relates to the instantaneous rate of change and the slope of the tangent line. It also covers finding the equation of a tangent line, and identifying horizontal and vertical tangents.

Derivatives and Tangents

Table of Contents

  1. What is the Derivative of a Function?
  2. How are Derivatives and Tangents Related?
  3. Finding the Equation of a Tangent to a Curve
  4. Horizontal and Vertical Tangents
  5. Worked Example
  6. Practice Questions
  7. Glossary
  8. Summary and Key Takeaways

What is the Derivative of a Function?

The derivative of a function describes the instantaneous rate of change of the function at any given point. It is equal to the slope of the curve at that point.

Key Concept

The derivative of the function ff is defined by:

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

This definition is valid for values of xx where this limit exists. Note that the derivative is also a function of xx.

There are several ways to denote the derivative of f(x)f(x):

  • f′(x)f'(x)
  • dydx\frac{dy}{dx}
  • y′y'

You may also see "the derivative of ..." written as " ... differentiated". They mean the same thing.

The value of the derivative of a function at a point is equal to the slope of the tangent to the graph at that point.

Key Concept

A tangent to a curve is a line that just touches the curve at one point but doesn't cut it at or near that point. How...

Previous Topic - Instantaneous Rate of ChangeNext Topic - Estimating the Derivative at a Point

Question 1 of 9

Which of the following represents the limit definition of the derivative of a function f(x)f(x)? 🤔

f′(x)=lim⁡h→0f(x)−f(x+h)hf'(x) = \lim_{h \to 0} \frac{f(x) - f(x+h)}{h}

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

f′(x)=lim⁡h→∞f(x+h)−f(x)hf'(x) = \lim_{h \to \infty} \frac{f(x+h) - f(x)}{h}

f′(x)=lim⁡h→0f(x+h)+f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) + f(x)}{h}