Differentiating Implicitly


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


xy = cot(xy)

Verified step by step guidance

1. Start by recognizing that the equation xy = cot(xy) involves both x and y, and y is a function of x. This means we need to use implicit differentiation.
2. Differentiate both sides of the equation with respect to x. For the left side, use the product rule: d/dx[xy] = x(dy/dx) + y.
3. For the right side, differentiate cot(xy) with respect to x. Use the chain rule: d/dx[cot(u)] = -csc^2(u) * du/dx, where u = xy. So, differentiate xy with respect to x, which gives: d/dx[xy] = x(dy/dx) + y.
4. Set the derivatives equal: x(dy/dx) + y = -csc^2(xy) * (x(dy/dx) + y).
5. Solve for dy/dx by isolating it on one side of the equation. This involves algebraic manipulation to factor out dy/dx and simplify the expression.

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 as implicit functions of x. This method is essential when dealing with equations like xy = cot(xy), where y is not 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 crucial when differentiating terms involving products or compositions of x and y, such as cot(xy).

Trigonometric Derivatives

Understanding the derivatives of trigonometric functions is vital for solving problems involving implicit differentiation. For example, the derivative of cot(u) with respect to u is -csc^2(u). When differentiating an equation like xy = cot(xy), knowing these derivatives allows you to correctly apply the chain rule and find dy/dx, especially when trigonometric functions are involved.