Find derivatives using the quotient rule II

Learning Objectives Apply the quotient rule to functions involving trigonometric expressions. Apply the quotient rule to functions involving exponential and natural logarithmic expressions. Combine the quotient rule with the chain rule to differentiate complex composite functions. Combine the quotient rule with the product rule when differentiating complex rational functions. Algebraically simplify the resulting derivative expression after applying the quotient rule. Find the equation of a tangent line to a rational function at a specific point. How does a GPS satellite calculate its changing velocity relative to a ground station, a calculation involving complex ratios? 🛰️ Let's find out how to handle the derivatives of these advanced functions! This tutorial builds up...

TermDefinitionExample Quotient RuleA formula used to find the derivative of a function that is the ratio of two other differentiable functions.To find the derivative of f(x) = sin(x) / x, we use the quotient rule. Chain RuleA formula used to find the derivative of a composite function (a function within a function).To find the derivative of g(x) = (2x + 1)^3, we use the chain rule, where the 'outer' function is u^3 and the 'inner' function is 2x + 1. Composite FunctionA function created by applying one function to the results of another function.In h(x) = e^(x^2), the function is a composition of f(u) = e^u and g(x) = x^2. Derivative of Trigonometric FunctionsThe rates of change for trigonometric functions.The derivative of sin(x) is cos(x), and the derivative of tan(x...

The Quotient Rule If f(x) = u(x) / v(x), then f'(x) = [v(x)u'(x) - u(x)v'(x)] / [v(x)]^2 Use this rule when you need to differentiate a function that is a fraction or ratio of two other functions. A common mnemonic is 'low d-high minus high d-low, over the square of what's below'. The Chain Rule If h(x) = f(g(x)), then h'(x) = f'(g(x)) * g'(x) Use this rule when differentiating a composite function. You differentiate the 'outside' function while keeping the 'inside' function the same, then multiply by the derivative of the 'inside' function.