Differentiation Formulas

The Essential Guide to Differentiation: Functions, Formulas, and Applications

What is Differentiation?

Differentiation is a fundamental concept in calculus that involves finding the rate of change of a function with respect to its input variable. It provides valuable insights into the behavior and characteristics of functions.

Functions and Their Derivatives

Consider the following functions:

• WEB Example 1:
• F(x) = 15x¹⁰⁰ + 3x¹² + 5x⁴⁶
• f(x) = 15x + 100 + 3x + 12 + 5x + 46
• G(t) = 2t⁶ + 7t⁶
• g(t) = 2t + 6 + 7t + 6

The derivative of each function represents its instantaneous rate of change at a given point.

General Differentiation Formulas

Some general differentiation formulas include:

• Power Rule: d/dx(xⁿ) = nx^n-1
• Constant Rule: d/dx(c) = 0
• Sum Rule: d/dx(f(x) + g(x)) = f'(x) + g'(x)
• Difference Rule: d/dx(f(x) - g(x)) = f'(x) - g'(x)
• Constant Multiple Rule: d/dx(cf(x)) = c*f'(x)

Applications of Differentiation

Differentiation has numerous applications in various fields, including:

• Calculating the slope of a tangent line
• Finding the extrema (maximum and minimum points) of a function
• Solving optimization problems
• Modeling growth and decay rates