Introduction to Calculus

What is Calculus?

Calculus is a branch of mathematics that studies how things change. It provides tools to understand and describe the motion and change in the world around us. The two main branches of calculus are:

• Differentiation: This is about finding the rate at which one quantity changes with respect to another. It's like finding the slope of a curve at any point.
• Integration: This is about finding the total size or value, such as areas under curves.

Differentiation of Algebraic Functions

Differentiation involves finding the derivative of a function. The derivative represents the rate of change of a function as its input changes. For algebraic functions, the derivative can be found using basic rules.

Basic Differentiation Rules

• Power Rule: If $f(x) = x^n$, then the derivative $f'(x) = nx^{n-1}$.
• Constant Rule: The derivative of a constant is zero.
• Sum Rule: The derivative of a sum is the sum of the derivatives.

Worked Example

Example

Example 1: Differentiate $f(x) = 4x^3 - 2x + 7$

Solution:

• For $4x^3$, the derivative is $4 \times 3x^{3-1} = 12x^2$.
• For $-2x$, the derivative is $-2 \times 1x^{1-1} = -2$.
• For $7$, the derivative is 0 (constant rule).

Thus, $f'(x) = 12x^2 - 2$.