Use implicit differentiation to find dy/dx.
sin xy = x+y

Verified step by step guidance

1. Start by differentiating both sides of the equation with respect to x. The equation is sin(xy) = x + y.
2. Apply the chain rule to differentiate sin(xy). The derivative of sin(u) with respect to u is cos(u), so differentiate sin(xy) to get cos(xy) * (d/dx)(xy).
3. Use the product rule to differentiate xy with respect to x. The product rule states that d(uv)/dx = u(dv/dx) + v(du/dx). Here, u = x and v = y, so the derivative is y + x(dy/dx).
4. Substitute the result from the product rule into the chain rule expression: cos(xy) * (y + x(dy/dx)).
5. Differentiate the right side of the equation, x + y, with respect to x. The derivative of x is 1, and the derivative of y with respect to x is dy/dx. Set the derivatives equal: cos(xy) * (y + x(dy/dx)) = 1 + dy/dx. Solve for dy/dx.

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. Instead of solving for one variable in terms of the other, we differentiate both sides of the equation with respect to the independent variable, applying the chain rule as necessary. This method is particularly useful for equations involving products or compositions of functions.

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. It states that if a function y is defined as a function of u, which in turn is a function of x, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This is essential in implicit differentiation when dealing with products of variables.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental functions in calculus that relate angles to ratios of sides in right triangles. In the context of implicit differentiation, understanding how to differentiate these functions is crucial, as they often appear in equations involving angles and products of variables. For example, the derivative of sin(u) is cos(u) multiplied by the derivative of u, which is applied when differentiating equations like sin(xy).