Quotient rule

Learning Objectives State the quotient rule formula from memory. Identify when the quotient rule is the appropriate method for finding a derivative. Apply the quotient rule to find the derivatives of rational functions, including those with polynomial, trigonometric, and exponential components. Correctly simplify the algebraic expressions that result from applying the quotient rule. Use the quotient rule to solve problems, such as finding the equation of a tangent line to a curve at a given point. Distinguish between the quotient rule and the product rule, and avoid common errors in their application. How does a company find the exact rate at which its average cost per item is changing at a specific production level? 🤔 The quotient rule gives us the answer! This tutorial f...

TermDefinitionExample 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.The derivative of f(x) = x^2 is f'(x) = 2x. At x=3, the slope of the tangent line is 2(3) = 6. QuotientA quotient is the result obtained by dividing one quantity by another.In the function h(x) = sin(x) / x, the expression is a quotient. Rational FunctionA function that can be written as the ratio of two polynomial functions, where the denominator is not the zero polynomial.f(x) = (3x^2 + 2x - 1) / (x + 5) is a rational function. NumeratorThe expression written above the line in a fraction, representing the dividend.In f(x)...

The Quotient Rule If h(x) = \frac{f(x)}{g(x)}, then h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} Use this rule when you need to find the derivative of a function that is structured as one differentiable function divided by another. A common mnemonic is 'Low D-High minus High D-Low, over the square of what's below', where 'Low' is g(x) and 'High' is f(x). The Power Rule \frac{d}{dx} (x^n) = nx^{n-1} This is a fundamental rule used to find the derivative of a variable raised to a constant power. It is frequently used to find the derivatives of the numerator and denominator functions before applying the quotient rule.