Product rule

Learning Objectives State the Product Rule formula using both Lagrange (f'(x)) and Leibniz (dy/dx) notation. Identify when the Product Rule is the appropriate method for finding a derivative. Correctly apply the Product Rule to find the derivatives of functions involving products of polynomials. Apply the Product Rule to functions involving products of trigonometric, exponential, and logarithmic functions. Combine the Product Rule with other differentiation rules, such as the Power Rule and Chain Rule. Simplify the resulting derivative expression after applying the Product Rule. Use the Product Rule to solve problems, such as finding the equation of a tangent line at a specific point. How do you find the rate of change of a company's revenue when both the price o...

TermDefinitionExample DerivativeThe derivative of a function, denoted f'(x) or dy/dx, represents the instantaneous rate of change of the function with respect to its variable. Geometrically, it gives the slope of the tangent line to the function's graph at any point.The derivative of f(x) = x^2 is f'(x) = 2x. Differentiable FunctionA function is differentiable at a point if its derivative exists at that point. This means the function's graph is smooth and has no sharp corners, cusps, or vertical tangents.f(x) = x^3 is differentiable everywhere. g(x) = |x| is not differentiable at x=0. Product of FunctionsA new function, h(x), created by multiplying two (or more) functions, f(x) and g(x), together.If f(x) = x^2 and g(x) = sin(x), their product is h(x) = x^2 * sin(x). La...

The Product Rule (Lagrange Notation) If h(x) = f(x)g(x), then h'(x) = f'(x)g(x) + f(x)g'(x) Use this formula when you have a function that is one function multiplied by another. It states: 'The derivative of the first function times the second, plus the first function times the derivative of the second.' The Product Rule (Leibniz Notation) If y = uv, where u and v are functions of x, then \frac{dy}{dx} = v\frac{du}{dx} + u\frac{dv}{dx} This is an alternative way to write the Product Rule, often useful in related rates or implicit differentiation problems. It represents the same calculation as the Lagrange form.