In Exercises 27–32, find dy/dx.
ln y = e^y sinx

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Start with the given equation: \(\ln y = e^{y} \sin x\).

Differentiate both sides with respect to \(x\). Remember that \(y\) is a function of \(x\), so use implicit differentiation.

For the left side, use the chain rule: \(\frac{d}{dx}(\ln y) = \frac{1}{y} \frac{dy}{dx}\).

For the right side, use the product rule: \(\frac{d}{dx}(e^{y} \sin x) = e^{y} \cos x + \sin x \cdot e^{y} \frac{dy}{dx}\) (note that \(\frac{d}{dx}(e^{y}) = e^{y} \frac{dy}{dx}\)).

Set up the equation: \(\frac{1}{y} \frac{dy}{dx} = e^{y} \cos x + \sin x \cdot e^{y} \frac{dy}{dx}\), 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 rather than explicitly as a function of x. It involves differentiating both sides of an equation with respect to x, treating y as a function of x and applying the chain rule to terms involving y.

Chain Rule

The chain rule is a method for differentiating composite functions. When differentiating expressions like ln(y) or e^y, where y is a function of x, the chain rule requires multiplying the derivative of the outer function by the derivative of the inner function (dy/dx).

Differentiation of Exponential and Logarithmic Functions

Understanding how to differentiate exponential functions like e^y and logarithmic functions like ln(y) is essential. The derivative of ln(y) is (1/y) dy/dx, and the derivative of e^y is e^y dy/dx, both requiring the chain rule when y depends on x.

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