Differentiation

Introduction to Differentiation

• Differentiation is a fundamental concept in calculus used to analyse rates of change.
• It involves finding the derivative of a function.
• The derivative represents the rate of change of one quantity with respect to another.

Basic Differentiation Rules

• The derivative of a constant is zero.
• The Power Rule: The derivative of x^n, where n is any real number, is n*x^(n-1).
• Product Rule: The derivative of the product of two functions (fg)’ = f’g + fg’.
• Quotient Rule: The derivative of the quotient of two functions (f/g)’= (f’g - fg’)/g^2.
• Chain Rule: Used to differentiate a composition of functions.

Differentiating Common Functions

• The derivative of sin(x) is cos(x).
• The derivative of cos(x) is -sin(x).
• The derivative of tan(x) is sec^2(x).
• The derivative of e^x is e^x.
• The derivative of ln(x) is 1/x.

Applications of Differentiation

• Gradient on a Curve: Derivatives can be used to find the slope of a function at a particular point.
• Local Minima and Maxima: The derivative of a function is zero at local minima and maxima.
• Rates of Change: Derivatives can determine rates of change in real-world applications, such as velocity and acceleration.

Higher-Order Derivatives

• Second, third and higher-order derivatives are found by repeating the differentiation process.
• In physics, the second derivative of displacement with respect to time gives acceleration.
• In economics, the second derivative is used to find points of diminishing returns.