Find tangent lines using implicit differentiation

Learning Objectives Differentiate an equation implicitly with respect to x. Correctly apply the chain rule to terms involving y when differentiating with respect to x. Algebraically solve for the derivative, dy/dx, after implicit differentiation. Evaluate the derivative at a specific point (x, y) to find the slope of the tangent line. Write the equation of a tangent line to an implicitly defined curve at a given point using the point-slope form. Identify points where a curve has a horizontal or vertical tangent line by analyzing the derivative dy/dx. How do you find the slope on a curve like a circle, which isn't a standard function? 🤔 Implicit differentiation is the key! This tutorial will teach you how to find the equation of a tangent line for curves that can'...

TermDefinitionExample Implicit RelationAn equation involving two or more variables (like x and y) where one variable is not explicitly solved for in terms of the other.The equation of a circle, x² + y² = 25, is an implicit relation. It's difficult to write it as a single function y = f(x). Explicit FunctionA function where the dependent variable (usually y) is isolated on one side of the equation, explicitly defining it in terms of the independent variable (x).y = 3x² - 2x + 5 is an explicit function. Implicit DifferentiationThe process of finding the derivative of an implicit relation by differentiating both sides of the equation with respect to a variable (usually x), treating the other variable (y) as a differentiable function of x.Differentiating x² + y² = 25 with respect to x yi...

Chain Rule for Implicit Differentiation \frac{d}{dx}[f(y)] = f'(y) \cdot \frac{dy}{dx} This is the most critical rule. Whenever you differentiate a function of y with respect to x, you first take the derivative with respect to y, and then multiply by the derivative of y with respect to x (which is dy/dx). Product Rule with Implicit Differentiation \frac{d}{dx}(x \cdot y) = (1 \cdot y) + (x \cdot \frac{dy}{dx}) When differentiating a product of x and y, you must apply the standard product rule. Remember that the derivative of y with respect to x is dy/dx. Point-Slope Form of a Line y - y_1 = m(x - x_1) After finding the slope (m) by evaluating dy/dx at the point (x₁, y₁), use this formula to write the final equation of the tangent line.