Find derivatives using the product rule I

Learning Objectives State the product rule formula from memory. Identify when the product rule is necessary to find a derivative. Decompose a function into its two component functions for use in the product rule. Apply the product rule to find the derivative of a function that is the product of two polynomials. Apply the product rule to find the derivative of a function involving a polynomial and a basic trigonometric function (sin(x), cos(x)). Simplify the resulting derivative expression after applying the product rule. How do you find the rate of change of a company's revenue if both the price of an item and the quantity sold are changing over time? 📈 Welcome to the Product Rule! This is a fundamental tool in calculus for finding the derivative of a function that is...

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.If f(x) = x^2, its derivative is f'(x) = 2x. Product of FunctionsA new function, let's call it h(x), that is formed by multiplying two other functions, f(x) and g(x), together.If f(x) = x^3 and g(x) = sin(x), their product is h(x) = x^3 * sin(x). Differentiable FunctionA function whose derivative exists at each point in its domain. Most functions you've encountered, like polynomials and basic trig functions, are differentiable everywhere.f(x) = 5x^4 - 2x is a differentiable function. Power RuleA fundamental differ...

The Product Rule If h(x) = f(x)g(x), then h'(x) = f'(x)g(x) + f(x)g'(x). Use this rule when you need to find the derivative of a function that is the product of two other differentiable functions. The pattern is: 'the derivative of the first function times the second function, plus the first function times the derivative of the second function'. The Product Rule (Leibniz Notation) d/dx[uv] = v(du/dx) + u(dv/dx) This is an alternative way to write the product rule, often used when the functions are represented by variables like u and v. It represents the same process. The Power Rule d/dx[x^n] = nx^(n-1) This is a prerequisite rule. You will use it to find the derivatives of the individual polynomial functions within the product rule.