Derivatives

A derivative is a measure of how a function changes as its input changes. It represents the rate of change of the function at a particular point. In calculus, the derivative of a function is denoted by f'(x) or dy/dx, and it tells us how the function's output changes with respect to its input.

Calculating Derivatives

There are several methods for calculating derivatives, including:

• Using the power rule: If f(x) = xⁿ, then f'(x) = nx^n-1
• Using the product rule: If f(x) = u(x)v(x), then f'(x) = u'v + uv'
• Using the chain rule: If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x)

Applications of Derivatives

Derivatives have many applications in real-world problems, including:

• Finding maximum and minimum values of functions
• Modeling rates of change in physics and engineering
• Optimizing functions in economics and business

Study Guide

When studying derivatives, it's important to:

1. Understand the basic rules for finding derivatives, such as the power rule, product rule, and chain rule
2. Practice solving problems involving derivatives to gain fluency in the techniques
3. Apply derivatives to real-world scenarios to understand their practical significance

Remember to always check your understanding by solving problems and seeking help when needed. Understanding derivatives is crucial for mastering calculus and its applications.