Find derivatives using implicit differentiation

Learning Objectives Define the difference between an explicit and an implicit function. Identify when implicit differentiation is necessary or advantageous. Correctly apply the chain rule to differentiate terms involving the variable y with respect to x. Differentiate both sides of an equation with respect to x, using all standard differentiation rules. Algebraically isolate the term dy/dx to find the derivative of an implicit function. Calculate the slope of a tangent line to an implicit curve at a specific point. How can you find the slope of a tangent line to a circle, an equation like x² + y² = 25, which isn't even a function? 🤔 This tutorial introduces implicit differentiation, a powerful technique for finding derivatives when you can't easily solve for &#03...

TermDefinitionExample Explicit FunctionA function where the dependent variable (usually 'y') is explicitly isolated on one side of the equation, written in the form y = f(x).y = 3x^2 - sin(x) Implicit FunctionA relationship between variables (usually 'x' and 'y') where the dependent variable is not isolated on one side of the equation.x^2 + y^2 = 25 The Operator d/dxA mathematical instruction that means 'take the derivative of the following expression with respect to the variable x'.\frac{d}{dx}(x^3) = 3x^2 The Chain Rule for Implicit DifferentiationWhen differentiating a term containing 'y' with respect to 'x', you first differentiate the term with respect to 'y' and then multiply by dy/dx.\frac{d}{dx}(y^3) = 3y^2 \cdo...

The Fundamental Step \frac{d}{dx}[LHS] = \frac{d}{dx}[RHS] To begin, take the derivative of every single term on both the Left-Hand Side (LHS) and Right-Hand Side (RHS) of the equation with respect to x. Chain Rule for Functions of y \frac{d}{dx}[f(y)] = f'(y) \cdot \frac{dy}{dx} This is the most critical rule. Whenever you differentiate a term involving y, you must multiply the result by dy/dx. This is because y is treated as a function of x. Product Rule with Implicit Terms \frac{d}{dx}[x \cdot y] = (1 \cdot y) + (x \cdot 1 \cdot \frac{dy}{dx}) = y + x \frac{dy}{dx} When a term is a product of x and y, you must apply the product rule. Remember to use the chain rule when differentiating the y part.