Find derivatives using the quotient rule I

Learning Objectives State the quotient rule formula from memory. Identify when the quotient rule is the appropriate method for finding a derivative. Correctly identify the numerator function and the denominator function. Accurately calculate the derivatives of the numerator and denominator functions. Apply the quotient rule to find the derivative of rational and other quotient functions. Simplify the resulting derivative expression using algebraic manipulation. How do you find the rate of change of a ratio, like a city's population density or an athlete's performance average? 🤔 The quotient rule gives us the answer! This tutorial introduces the quotient rule, a fundamental tool in calculus for finding the derivative of a function that is the division of two other...

TermDefinitionExample DerivativeThe derivative of a function measures the instantaneous rate of change of the function with respect to one of its variables. It represents the slope of the tangent line to the function's graph at a specific point.If f(x) = x^3, its derivative is f'(x) = 3x^2. QuotientThe result obtained by dividing one quantity by another.In the function h(x) = (x+1)/(x-1), the expression (x+1)/(x-1) is a quotient. Rational FunctionA function that can be expressed as the ratio of two polynomial functions, P(x)/Q(x), where Q(x) is not the zero polynomial.f(x) = (3x^2 - 5x + 2) / (x + 4) is a rational function. Numerator FunctionIn a quotient of functions, f(x)/g(x), the function in the top part of the fraction, f(x), is the numerator function.In h(x) = sin(x) / x^2...

The Quotient Rule If h(x) = f(x) / g(x), then h'(x) = [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2 Use this formula to find the derivative of any function that is structured as one differentiable function divided by another. Remember the denominator must not be zero. The Power Rule d/dx(x^n) = nx^(n-1) A fundamental rule used to find the derivative of a variable raised to a constant power. You will frequently use this to find the derivatives of the numerator and denominator functions if they are polynomials.