Implicit differentiation

Learning objectives

Differentiate both sides of an equation that defines y implicitly as a function of x, applying the chain rule to every term containing y, and solve for dy/dx

Find the slope of, and the equation of, the tangent line to an implicitly defined curve at a given point, verifying that the point lies on the curve

Find the points on an implicit curve where the tangent line is horizontal or vertical by setting the numerator or the denominator of dy/dx equal to zero and solving simultaneously with the original equation

Find the second derivative d²y/dx² of an implicit relation, substituting the expression for dy/dx into the result and simplifying using the original equation