Understanding Derivatives in Algebra

Derivatives measure how a function changes as its input changes. They are a key concept in calculus and algebra.

What is a Derivative?

A derivative represents the rate of change or the slope of a function at a point.

It is denoted as \( f'(x) \) or \( \frac{dy}{dx} \).

Basic Derivative Rules

• \( \frac{d}{dx} x^n = nx^{n-1} \)
• \( \frac{d}{dx} e^x = e^x \)
• \( \frac{d}{dx} \sin x = \cos x \)

Applications of Derivatives

Derivatives are used to find slopes, rates of change, maxima, minima, and solve real-world problems.

Practice Problems

1. Find \( \frac{d}{dx} (3x^2 + 2x) \).
2. Find the slope of \( y = x^2 \) at \( x = 2 \).
3. Find \( \frac{d}{dx} (\sin x) \).