Chain Rule

The chain rule is a fundamental concept in calculus that allows us to find the derivative of a composite function. In other words, it helps us determine the rate of change of a function within another function. The chain rule is essential in calculus and is used extensively in the study of derivatives.

Formula

The chain rule can be expressed as:

\[ \frac{d}{dx} (f(g(x))) = f'(g(x)) \cdot g'(x) \]

Explanation

When we have a composite function, where one function is nested within another, the chain rule tells us how to find the derivative of the outer function with respect to the inner function, and then multiply it by the derivative of the inner function with respect to the variable. In simpler terms, it helps us find the derivative of a function within a function.

Example

Let's take the function \( f(x) = (3x^2 + 2x)^3 \). To find the derivative of this function with respect to x, we can use the chain rule as follows:

\[ \frac{d}{dx} (3x^2 + 2x) = 6x + 2 \]\[ \frac{d}{dx} (3x^2 + 2x)^3 = 3(3x^2 + 2x)^2 \cdot (6x + 2) \]

Study Guide

When studying the chain rule, it's important to understand the following key points:

Understand the concept of composite functions and how one function can be nested within another.
Memorize the chain rule formula and understand how it applies to finding derivatives of composite functions.
Practice applying the chain rule to various functions to gain proficiency in using the formula.
Recognize when to use the chain rule in finding derivatives, especially when dealing with complex functions.

It's also helpful to work through numerous examples and exercises to solidify your understanding of the chain rule and its application in calculus.

Remember that the chain rule is a powerful tool in calculus and is essential for finding derivatives of composite functions. With practice and understanding, you can master this concept and apply it to solve more complex problems in calculus.

◂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]