Derivatives Study Guide

What is a Derivative?

A derivative represents the rate of change of a function at a particular point. It measures how a function's output changes with respect to its input. In other words, it gives us the slope of the function at a specific point.

Notation

The derivative of a function f(x) is denoted by f'(x) or dy/dx. It can also be written as d/dx[f(x)].

Derivative Rules

There are several rules for finding derivatives, including:

• Power Rule: If f(x) = x^n, then f'(x) = nx^(n-1)
• Product Rule: If f(x) = u(x)v(x), then f'(x) = u'v + uv'
• Quotient Rule: If f(x) = u(x)/v(x), then f'(x) = (u'v - uv')/v^2
• Chain Rule: If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x)

Common Derivatives

Some common derivatives to remember include:

• f(x) = sin(x), f'(x) = cos(x)
• f(x) = cos(x), f'(x) = -sin(x)
• f(x) = e^x, f'(x) = e^x
• f(x) = ln(x), f'(x) = 1/x

Applications

Derivatives have many applications in real-world problems, including finding maximum and minimum values, analyzing motion, and solving optimization problems.

Study Tips

To study derivatives effectively, consider the following tips:

• Practice finding derivatives using various rules and techniques.
• Memorize common derivatives and their corresponding functions.
• Work on real-world applications of derivatives to understand their practical significance.
• Seek help from a tutor or study group if you encounter difficulties.