In Exercises 27–32, find dy/dx.
3+siny = y-x^3

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Start with the given equation: \(3 + \sin y = y - x^{3}\).

Differentiate both sides of the equation with respect to \(x\). Remember that \(y\) is a function of \(x\), so when differentiating terms involving \(y\), use the chain rule.

The derivative of the left side is \(\frac{d}{dx}(3) + \frac{d}{dx}(\sin y) = 0 + \cos y \cdot \frac{dy}{dx}\), since \(\frac{d}{dx}(3) = 0\) and \(\frac{d}{dx}(\sin y) = \cos y \cdot \frac{dy}{dx}\) by the chain rule.

The derivative of the right side is \(\frac{d}{dx}(y) - \frac{d}{dx}(x^{3}) = \frac{dy}{dx} - 3x^{2}\).

Set the derivatives equal: \(\cos y \cdot \frac{dy}{dx} = \frac{dy}{dx} - 3x^{2}\). Then, collect all terms involving \(\frac{dy}{dx}\) on one side and factor it out to solve for \(\frac{dy}{dx}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Implicit Differentiation

Implicit differentiation is used when y is defined implicitly by an equation involving both x and y. Instead of solving for y explicitly, we differentiate both sides with respect to x, treating y as a function of x and applying the chain rule when differentiating terms involving y.

Chain Rule

The chain rule is a method for differentiating composite functions. When differentiating terms like sin(y), where y is a function of x, we first differentiate the outer function (sin) with respect to y, then multiply by dy/dx, the derivative of the inner function.

Solving for dy/dx

After differentiating both sides implicitly, dy/dx terms appear on both sides of the equation. The next step is to collect all dy/dx terms on one side and factor them out, then isolate dy/dx by dividing through by the remaining coefficient to find the derivative explicitly.

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