Differential Calculus is the subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus.
1. The Concept of the Derivative
The primary object of study in differential calculus is the derivative. The derivative of a function represents the rate of change of the function's output with respect to its input.
Visually, the derivative at any point on a graph is the slope of the tangent line to the curve at that point.
[Image of tangent line slope]If y = f(x), the derivative is written as:
f'(x) or dy/dx
2. Rules of Differentiation
We don't always have to use complex limits to find a derivative. Over time, mathematicians have discovered shortcuts known as differentiation rules.
A. The Power Rule
This is the most common rule. To differentiate x to a power, bring the power down in front and subtract one from the exponent.
Example: f(x) = x³ → f'(x) = 3x²
B. The Product Rule
Used when multiplying two functions together.
C. The Chain Rule
Used for composite functions (a function inside another function). It states that you take the derivative of the outside function and multiply it by the derivative of the inside function.
h'(x) = f'(g(x)) · g'(x)
3. Applications of Differential Calculus
Why do we care about derivatives? Because they describe the physical world.
- Physics: If you have a function for position, the derivative is velocity. The derivative of velocity is acceleration.
- Optimization: Derivatives help us find maximums and minimums. Business owners use this to maximize profit and minimize cost.
- Biology: Measuring the rate of population growth of a bacteria culture over time.
4. Higher Order Derivatives
You can take the derivative of a derivative! This is called the second derivative, written as f''(x). This tells us how the rate of change itself is changing (concavity).