Find derivatives of rational functions

Learning Objectives Identify a rational function and its component polynomial functions. State and correctly write the Quotient Rule from memory. Apply the Quotient Rule to find the derivative of various rational functions. Simplify the resulting derivative expression using algebraic manipulation. Find the derivative of rational functions where the numerator or denominator is a constant. Calculate the slope of the tangent line to a rational function at a specific point. How does a company find the exact point where its average cost per item is at its absolute minimum? 📈 Calculus gives us the tools to pinpoint that precise moment! This tutorial will focus on a crucial technique for finding the derivative of a rational function, which is a function expressed as a fraction of...

TermDefinitionExample Rational FunctionA function that can be written as the ratio of two polynomial functions, P(x) and Q(x), in the form f(x) = P(x) / Q(x), where Q(x) is not the zero polynomial.f(x) = (3x^2 + 2) / (x - 5) DerivativeThe derivative of a function measures the instantaneous rate of change of the function with respect to one of its variables. Geometrically, it represents the slope of the tangent line to the function's graph at a specific point.If f(x) = x^2, its derivative is f'(x) = 2x. Quotient RuleA formula used to find the derivative of a function that is the division (quotient) of two other differentiable functions.To find the derivative of h(x) = f(x) / g(x), we use the Quotient Rule. Polynomial FunctionAn expression consisting of variables and coefficients,...

The Quotient Rule If h(x) = f(x) / g(x), then h'(x) = [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2 This is the primary rule for differentiating rational functions. It involves the derivative of the top function, the original bottom function, the original top function, the derivative of the bottom function, and the square of the bottom function. The Power Rule d/dx [x^n] = n * x^(n-1) This rule is used to find the derivatives of the individual polynomial terms in the numerator and denominator before applying the Quotient Rule. The Constant Rule d/dx [c] = 0 The derivative of any constant is zero. This is important when a term in the numerator or denominator is a constant.