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)

Review the fundamental concepts of calculus, including limits, implicit differentiation, and applications of derivatives in optimization, physics, economics, and computer science. Learn the rules of differentiation, such as the power rule, product rule, quotient rule, and chain rule.