Mathematics Foundations/9.2 Derivatives and Differentiation Rules

9.2 Derivatives and Differentiation Rules Introduction The derivative is a fundamental concept in calculus that measures the rate at which a function changes at a particular point. It represents the instantaneous rate of change of a function with respect to one of its variables. The Derivative Concept The derivative of a function at a point is defined as the limit: Alternatively, using the limit notation with approaching : Geometric Interpretation Geometrically, the derivative represents the slope of the tangent line to the graph of at the point . Notation for Derivatives There are several notations used to denote derivatives: 1. Lagrange's notation: , , 2. Leibniz's notation: , , 3. Newton's notation: , 4. Euler's notation: , , Differentiability A function is differentiable at if exists. For a function to be differentiable at a point, it must be continuous at that point. However, continuity does not guarantee differentiability. For example, the function is continuous at but not differentiable there. Basic Differentiation Rules The following are fundamental differentiation rules for common functions: 1. Constant Rule: If (constant), then 2. Power Rule: If , then 3. Exponential Rule: If , then 4. Natural Logarithm Rule: If , then for 5. Trigonometric Rules: * If , then * If , then * If , then 6. Inverse Trigonometric Rules: * If , then * If , then * If , then Differentiation Rules for Combined Functions The following rules apply when functions are combined: 1. Sum Rule: If , then 2. Difference Rule: If , then 3. Product Rule: If , then 4. Quotient Rule: If , then 5. Chain Rule: If , then Higher-Order Derivatives The second derivative of , denoted or , is the derivative of the first derivative. Similarly, higher-order derivatives can be obtained by continuing this process. Physical Interpretation First derivative: rate of change (velocity for position functions) Second derivative: rate of change of the rate of change (acceleration for position functions) Implicit Differentiation When a function is defined implicitly by an equation (not solved explicitly for ), we can find by differentiating both sides of the equation with respect to and solving for . Example For the circle equation : Differentiating both sides with respect to : Solving for : Logarithmic Differentiation Logarithmic differentiation is useful for differentiating products, quotients, or expressions with variable exponents. The technique involves taking the natural logarithm of both sides of an equation and then differentiating. Example To find the derivative of : Taking the natural logarithm: Differentiating both sides: Solving for : Derivatives of Parametric Equations For a curve defined parametrically by and , the derivative is given by: provided that . Applications of Derivatives Derivatives have numerous applications, including: 1. Finding the equation of tangent and normal lines to curves 2. Analyzing the motion of objects (velocity and acceleration) 3. Calculating rates of change in various physical processes 4. Approximating function values using linear approximation 5. Optimization problems (finding maxima and minima) L'Hôpital's Rule L'Hôpital's Rule is used to evaluate limits that result in indeterminate forms such as or . If results in an indeterminate form, and and are differentiable with near , then: provided the limit on the right exists or is infinite. Exercises 1. Find the derivative of using the product rule. 2. Use implicit differentiation to find for the equation . 3. Find the second derivative of . 4. Use logarithmic differentiation to find the derivative of . 5. Evaluate using L'Hôpital's Rule. 6. Find the equation of the tangent line to the curve at the point where . 7. For the parametric curve , , find at .