Calculus: Differentiation Rules

Study Notes

Differentiation is a fundamental concept in calculus that deals with the rate of change of a function with respect to one of its variables.
It is a measure of how a function changes as its input changes.

Power Rule: If f(x) = x^n, then f'(x) = nx^(n-1)
Product Rule: If f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x)
Quotient Rule: If f(x) = u(x)/v(x), then f'(x) = (u'(x)v(x) - u(x)v'(x)) / v(x)^2
Chain Rule: If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x)

The derivative of a function at a point represents the slope of the tangent line to the graph of the function at that point.
The derivative can be used to find the maximum and minimum values of a function.

The second derivative of a function f(x) is denoted as f''(x) or (d^2/dx^2)f(x)
Higher-order derivatives can be used to find the concavity and inflection points of a function.

Optimization: Differentiation is used to find the maximum and minimum values of a function, which is crucial in many fields such as economics, physics, and engineering.
Physics: Differentiation is used to model the motion of objects, including the acceleration and velocity of particles and the curvature of space-time.
Economics: Differentiation is used to model the behavior of economic systems, including the impact of changes in supply and demand on prices and quantities.

Deals with the rate of change of a function with respect to one of its variables.
Measures how a function changes as its input changes.

Power Rule: If f(x) = x^n, then f'(x) = nx^(n-1).
Product Rule: If f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x).
Quotient Rule: If f(x) = u(x)/v(x), then f'(x) = (u'(x)v(x) - u(x)v'(x)) / v(x)^2.
Chain Rule: If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).

Derivative of a function at a point represents the slope of the tangent line to the graph of the function at that point.
Derivative can be used to find the maximum and minimum values of a function.

Second derivative of a function f(x) is denoted as f''(x) or (d^2/dx^2)f(x).
Higher-order derivatives can be used to find the concavity and inflection points of a function.

Optimization: Used to find the maximum and minimum values of a function, crucial in economics, physics, and engineering.
Physics: Used to model the motion of objects, including acceleration and velocity of particles and curvature of space-time.
Economics: Used to model the behavior of economic systems, including impact of changes in supply and demand on prices and quantities.