Differentiation

Basic Concepts

Differentiation is a process in calculus used to measure the rate at which a quantity is changing.
It involves finding the derivative of a function.
The derivative represents the instantaneous rate of change of the function - effectively the gradient of the function at any given point.

Power Rule

The power rule states that if f(x) = x^n, then the derivative of function f(x) is given by f’(x) = n*x^(n-1).
This rule allows us to differentiate monomials.

Product and Quotient Rules

The product rule and quotient rule allow differentiation of functions that are products or quotients of simpler functions.
Product rule: If f(x) = g(x)h(x), then f’(x) = g’(x)h(x) + g(x)*h’(x).
Quotient rule: If f(x) = g(x)/h(x), then f’(x) = (g’(x)h(x) - g(x)h’(x))/[h(x)]^2.

Chain Rule

The chain rule allows us to differentiate composite functions.
If a function y = f(u) is composed with another function u = g(x), the derivative of y with respect to x is d/dx[y] = d/du[f(u)]* du/dx.

Higher Derivatives

We can differentiate a function more than once to get the second derivative, third derivative, and so on.
These higher derivatives provide information about the rate of change of the gradient of the function.

Applications

Differentiation is used in multiple areas, including physics, engineering, economics, biology, and more.
For instance, it can be used to find maximum and minimum values of a function - useful for optimising quantities.

Remember: Differentiation provides us with a mathematical tool to analyse how a function is changing. Understanding differentiation is crucial to understand more advanced mathematics, including integration, differential equations, and multivariate calculus.