Introduction to Calculus: Differentiation

Introduction to Calculus

• Calculus is a branch of mathematics focused on change, encompassing differential and integral calculus.
• Differential calculus examines rates of change, curve slopes, and tangents.
• Integral calculus deals with accumulation, areas under curves, and volumes of solids.

Differentiation

• Differentiation finds the derivative, the instantaneous rate of change of a function at a point.
• The derivative shows the slope of the tangent line to the function's graph at that point.
• The derivative of function f(x) is denoted as f'(x) or dy/dx.

Basic Differentiation Rules

• Constant Rule: The derivative of a constant is zero. d/dx(c) = 0 (c is a constant)
• Power Rule: The derivative of xⁿ is nx^n-1. d/dx(xⁿ) = nx^n-1
• Constant Multiple Rule: The derivative of a constant times a function is the constant times the function's derivative. d/dx(cf(x)) = c * d/dx(f(x))
• Sum/Difference Rule: The derivative of a sum or difference of functions is the sum or difference of their derivatives. d/dx(f(x) ± g(x)) = f'(x) ± g'(x)
• Product Rule: The derivative of a product of two functions is the first function times the derivative of the second, plus the second function times the derivative of the first. d/dx(f(x) * g(x)) = f(x) * g'(x) + g(x) * f'(x)
• Quotient Rule: The derivative of a quotient is the denominator times the numerator's derivative minus the numerator times the denominator's derivative, all divided by the denominator squared. d/dx(f(x) / g(x)) = [g(x) * f'(x) - f(x) * g'(x)] / [g(x)]²

Derivatives of Common Functions

• Derivative of sin(x): cos(x)
• Derivative of cos(x): -sin(x)
• Derivative of e^x: e^x
• Derivative of ln(x): 1/x (x > 0)

Higher-Order Derivatives

• A derivative of a derivative is a higher-order derivative.
• The second derivative is f''(x) or d²y/dx², representing the rate of change of the rate of change.
• Higher-order derivatives are found by repeated application of differentiation rules.

Applications of Differentiation

• Finding tangent line slopes
• Locating critical points (local maxima and minima)
• Solving optimization problems (finding maximum or minimum values)
• Sketching curves
• Determining concavity and inflection points
• Calculating rates of change (including related rates problems)