Calculus 1: Derivatives and Their Applications
Understanding Derivatives
A derivative represents the rate at which a function is changing at any given point. It's a fundamental
concept in calculus, used for understanding dynamics, physics, and other sciences.
The derivative of a function f(x) at a point x is defined as:
f'(x) = lim (h -> 0) [(f(x + h) - f(x)) / h]
This formula calculates the slope of the tangent line to the function at point x.
Example:
Consider the function f(x) = x^2. We want to find the derivative f'(x).
Step 1: Apply the derivative formula
f'(x) = lim (h -> 0) [(x^2 + 2xh + h^2 - x^2) / h]
Step 2: Simplify the expression
f'(x) = lim (h -> 0) [2xh + h^2 / h]
Step 3: Further simplification
f'(x) = lim (h -> 0) [2x + h]
Step 4: Apply the limit
f'(x) = 2x
The derivative of f(x) = x^2 is f'(x) = 2x. This indicates that the slope of the tangent line to the curve
Calculus 1: Derivatives and Their Applications
at any point x is 2x.
Derivatives have various applications, including in physics for determining velocity and acceleration,
in economics to find marginal cost and revenue, and in engineering for modeling and analysis.