Calculus Derivatives

Derivatives

Definition

• The derivative of a function f(x) at a point x=a represents the rate of change of the function with respect to x at that point.
• Notation: f'(a) or (d/dx)f(x)|x=a

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)

Geometric Interpretation

• 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.

Higher-Order Derivatives

• The second derivative f''(x) represents the rate of change of the first derivative f'(x).
• Higher-order derivatives can be used to analyze the concavity and inflection points of a function.

Applications of Derivatives

• Optimization: Derivatives are used to find the maximum and minimum values of a function, which is crucial in many fields such as economics and physics.
• Physics: Derivatives are used to model the motion of objects, including the acceleration and velocity of particles and the curvature of space-time.
• Economics: Derivatives are used to model the behavior of economic systems, including the analysis of supply and demand curves.

Derivatives

Definition

• Derivative of a function f(x) at a point x=a represents the rate of change of the function with respect to x at that point.
• Notation: f'(a) or (d/dx)f(x)|x=a

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)

Geometric Interpretation

• 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.

Higher-Order Derivatives

• Second derivative f''(x) represents the rate of change of the first derivative f'(x).
• Higher-order derivatives can be used to analyze the concavity and inflection points of a function.

Applications of Derivatives

Optimization

• Derivatives are used to find the maximum and minimum values of a function, crucial in economics and physics.

Physics

• Derivatives are used to model the motion of objects, including acceleration and velocity of particles and curvature of space-time.

Economics

• Derivatives are used to model the behavior of economic systems, including analysis of supply and demand curves.