Differentiability

Differentiability is an important concept in calculus that deals with the smoothness of a function. A function is said to be differentiable at a point if it has a well-defined derivative at that point. The derivative of a function at a point represents the rate of change of the function at that point.

Definition

A function f(x) is said to be differentiable at a point c if the following limit exists:

f'(c) = lim_{x → c} (f(x) - f(c))/(x - c)

If this limit exists, then the function is said to be differentiable at point c. If the function is differentiable at every point in its domain, then it is called a differentiable function.

Derivative

The derivative of a function f(x) at a point c, denoted as f'(c), gives the slope of the tangent line to the graph of the function at that point. It represents the instantaneous rate of change of the function at that point.

Study Guide

Understand the concept of limits and continuity, as differentiability is closely related to these concepts.
Learn the definition of differentiability and how to calculate the derivative of a function at a point using the limit definition.
Practice finding the derivatives of various types of functions, including polynomials, trigonometric functions, exponential functions, and logarithmic functions.
Understand the connection between differentiability and the behavior of the function's graph, such as the existence of tangent lines and local extrema.
Study the differentiability of composite functions and understand the chain rule for finding the derivatives of composite functions.
Practice using differentiability to analyze the behavior of functions, such as determining where a function is increasing or decreasing, and finding the concavity of the graph.
Explore applications of differentiability in physics, engineering, economics, and other fields to understand its real-world significance.

Understanding the concept of differentiability is crucial for mastering calculus and its applications in various fields. Practice problems and seeking help from a tutor can help reinforce your understanding of this topic.

◂Math Worksheets and Study Guides Eighth Grade. Real numbers

The resources above cover the following skills:

The Number System

Know that there are numbers that are not rational, and approximate them by rational numbers.

Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. [8-NS1]

Expressions and Equations

Work with radicals and integer exponents.

Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. [8-EE2]