Calculus: Limits, Differentiation, and Applications

Limits Definition

• A derivative of a function f at a point x=a represents the rate of change of the function with respect to x at that point.
• The derivative is defined as a limit: f'(a) = lim(h → 0) [f(a + h) - f(a)]/h
• The limit is used to measure the instantaneous rate of change of the function.
• The derivative is a measure of how the function changes as its input changes.

Implicit Differentiation

• Implicit differentiation is a technique used to find the derivative of an implicitly defined function.
• The method involves differentiating both sides of the equation with respect to the variable, and then solving for the derivative.
• Chain rule is often used in implicit differentiation: dy/dx = dy/du * du/dx
• Implicit differentiation is useful when the function is not explicitly defined.

Applications of Derivatives

• Optimization: Derivatives are used to find the maximum or minimum values of a function.
• Physics: Derivatives are used to model real-world phenomena, such as velocity and acceleration.
• Economics: Derivatives are used to model economic systems, such as supply and demand curves.
• Computer Science: Derivatives are used in machine learning and artificial intelligence to optimize functions.

Rules of Differentiation

• 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)
• Sum and Difference Rule: If f(x) = u(x) ± v(x), then f'(x) = u'(x) ± v'(x)