Chain rule

Learning Objectives Identify the 'inner' and 'outer' functions of a composite function. State the chain rule using both function notation and Leibniz notation. Apply the chain rule to differentiate composite functions involving polynomial, trigonometric, exponential, and logarithmic expressions. Combine the chain rule with the product and quotient rules to find the derivatives of more complex functions. Solve problems involving rates of change that require the chain rule. Calculate higher-order derivatives of composite functions. If a spherical balloon is being inflated, how can we find the rate at which its radius is increasing at the exact moment we know the rate at which its volume is increasing? 🎈 The Chain Rule is a fundamental formula for differen...

TermDefinitionExample Composite FunctionA function that is formed by applying one function to the results of another function. It is often written as h(x) = f(g(x)).If f(x) = x^5 and g(x) = 2x + 3, the composite function is h(x) = f(g(x)) = (2x + 3)^5. Outer FunctionIn a composite function f(g(x)), the outer function is 'f', which acts on the inner function.For the function h(x) = sin(x^2), the outer function is f(u) = sin(u). Inner FunctionIn a composite function f(g(x)), the inner function is 'g', which is the input to the outer function.For the function h(x) = sin(x^2), the inner function is g(x) = x^2. DerivativeThe instantaneous rate of change of a function with respect to one of its variables, or the slope of the tangent line at a point on the function's gra...

The Chain Rule (Function Notation) If h(x) = f(g(x)), then h'(x) = f'(g(x)) \cdot g'(x) This form is often described as 'the derivative of the outside function (with the inside function left alone), times the derivative of the inside function'. The Chain Rule (Leibniz Notation) If y = f(u) and u = g(x), then \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} This notation is particularly useful for related rates problems and helps to visualize how the rates 'chain' together. The General Power Rule If y = [g(x)]^n, then \frac{dy}{dx} = n[g(x)]^{n-1} \cdot g'(x) This is a direct and very common application of the chain rule, used for any function raised to a power.