Differentiating Implicitly


Use implicit differentiation to find dy/dx in Exercises 1–14.


x²(x – y)² = x² – y²

Verified step by step guidance

1. Start by differentiating both sides of the equation with respect to x. The equation is x²(x – y)² = x² – y².
2. Apply the product rule to the left side of the equation. The product rule states that d(uv)/dx = u'v + uv', where u = x² and v = (x - y)².
3. Differentiate u = x² to get u' = 2x. Differentiate v = (x - y)² using the chain rule: v' = 2(x - y)(1 - dy/dx).
4. Apply the chain rule to the right side of the equation: differentiate x² to get 2x and differentiate -y² to get -2y(dy/dx).
5. Set the derivatives of both sides equal to each other and solve for dy/dx. This involves substituting the derivatives found in the previous steps and isolating dy/dx.

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function when it is not explicitly solved for one variable in terms of another. It involves differentiating both sides of an equation with respect to a variable, often x, while treating other variables, like y, as implicit functions of x. This method is essential when dealing with equations where y cannot be easily isolated.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. In implicit differentiation, the chain rule is often applied when differentiating terms involving y, as y is considered a function of x.

Product Rule

The product rule is a technique used to differentiate products of two or more functions. It states that the derivative of a product of two functions u(x) and v(x) is given by u'(x)v(x) + u(x)v'(x). In the given problem, the product rule is necessary to differentiate terms like x²(x – y)², where multiple functions of x are multiplied together.