Find derivatives using the product rule II

Learning Objectives Apply the product rule to functions involving trigonometric, exponential, and logarithmic components. Combine the product rule with the chain rule to differentiate composite functions. Differentiate functions that are the product of three or more functions. Calculate the second derivative of a function that requires the product rule. Find the equation of a tangent line to a curve where the derivative requires the product rule. Solve problems involving rates of change that require the product rule. How do you find the rate of change of a rocket's momentum when both its mass (burning fuel) and velocity are changing? 🤔 The product rule is the key! This tutorial builds on your existing knowledge of the product rule by tackling more complex functions. Y...

TermDefinitionExample Product Rule (Review)A formula used to find the derivative of a product of two or more functions.For h(x) = x² * sin(x), the derivative h'(x) is found by applying the product rule to f(x) = x² and g(x) = sin(x). Chain RuleA formula to compute the derivative of a composite function, which is a function formed by composing one function with another.To differentiate h(x) = cos(3x), we use the chain rule on the outer function cos(u) and the inner function u = 3x. Composite FunctionA function that is nested inside another function, written as f(g(x)).In the function y = e^(x²), the function g(x) = x² is the 'inner' function nested inside the 'outer' function f(u) = e^u. Higher-Order DerivativeThe result of differentiating a function more than once...

The Product Rule \frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x) Use this rule when you need to differentiate a function that is the product of two other functions. It is often stated as 'the derivative of the first times the second, plus the first times the derivative of the second'. The Chain Rule \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x) Use this rule when differentiating a composite function. Differentiate the 'outside' function while keeping the 'inside' function the same, then multiply by the derivative of the 'inside' function. The Product Rule for Three Functions \frac{d}{dx} [f(x)g(x)h(x)] = f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x) An extension of the product rule for a produ...