Definition of Differentiation

Sarah Miller

5 min read

Next Topic - Differentiability & Continuity

Listen to this study note

Study Guide Overview

This study guide covers estimating derivatives at a point using both graphs and tables. Key concepts include understanding that a derivative is the slope of the tangent line, and approximating this slope using nearby points. The guide emphasizes the importance of continuous and differentiable functions. Practice questions and a glossary are also included.

#Estimating Derivatives at a Point

#Estimating Derivatives Using a Graph

#How Can I Estimate a Derivative at a Point Using a Graph?

To estimate the derivative of a function at a particular point using its graph, follow these steps:

Identify the coordinates of the points that lie on the graph of the function.
Recall that the derivative of a function $f(x)$ at $x=a$ , denoted as $f'(a)$ , is equal to the slope of the tangent to the graph of $f(x)$ at $x=a$ .

Key Concept

To approximate the slope of the tangent to the graph of $f(x)$ at $x=a$ :

Find the slope of line segments joining nearby points that lie on the graph.
The function must be continuous and differentiable within the relevant interval for this method to be valid.

#Example

Consider the graph of $f(x)$ with points $A, B, C, D, E$ labeled with their coordinates. To approximate the derivative of $f(x)$ at $x=C$ , w...

How are we doing?

Give us your feedback and let us know how we can improve

Previous Topic - Derivatives & Tangents Next Topic - Differentiability & Continuity

Question 1 of 8

Let's warm up! 🚀 If $f(x)$ is a function and you want to estimate its derivative at $x=a$ using a graph, what are you essentially trying to find?

The y-coordinate of the point on the graph at $x=a$

The area under the curve of $f(x)$ from $0$ to $a$

The slope of the tangent to the graph of $f(x)$ at $x=a$

The x-intercept of the graph of $f(x)$