Problem 27 - Thomas Calculus

Question

find \(d y / d x.\) \begin{equation}\ln y=e^{y} \sin x\end{equation}

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = \frac{y e^y \cos x}{1 - y e^y \sin x} \)

Step 1: Apply Implicit Differentiation

Differentiate both sides of the equation \(\ln y = e^y \sin x\) with respect to \(x\). Remember that \(y\) is a function of \(x\), so apply the chain rule when differentiating terms involving \(y\). The derivative of \(\ln y\) is \((1/y)\frac{dy}{dx}\), and the derivative of \(e^y \sin x\) is \(e^y \sin x \frac{dy}{dx} + e^y \cos x\) using the product rule.

Step 2: Set the Derivatives Equal

After differentiating, write the equation: \( \frac{1}{y} \frac{dy}{dx} = e^y \sin x \frac{dy}{dx} + e^y \cos x \). Arrange this equation to collect all terms involving \(\frac{dy}{dx}\) on one side.

Step 3: Solve for \(\frac{dy}{dx}\)

Factor out \(\frac{dy}{dx}\) from the terms on one side: \( \frac{dy}{dx} \left( \frac{1}{y} - e^y \sin x \right) = e^y \cos x \). Solve for \(\frac{dy}{dx}\) by dividing both sides by \(\frac{1}{y} - e^y \sin x\): \( \frac{dy}{dx} = \frac{e^y \cos x}{\frac{1}{y} - e^y \sin x} \).

Step 4: Simplify the Expression

Simplify the expression for \(\frac{dy}{dx}\): Multiply both numerator and denominator by \(y\) to eliminate the fraction in the denominator: \( \frac{dy}{dx} = \frac{y e^y \cos x}{1 - y e^y \sin x} \). This is the simplest form of the derivative.

Key Concepts

Chain Rule

The chain rule is one of the essential techniques in calculus, particularly when dealing with composite functions. This rule helps us differentiate functions where one function is nested inside another. This situation arises frequently, especially in implicit differentiation.
In the given exercise, our function involves two parts:

• The term \( \ln y \) needs the chain rule because \( y \) is a function of \( x \), not \( x \) itself.

The chain rule essentially states that if you have a function \( g(y) \) and \( y = f(x) \), then the derivative \( \frac{d}{dx}[g(y)] = g'(y) \cdot \, \frac{dy}{dx} \).
When differentiating \( \ln y \), you apply the chain rule to get \( \frac{1}{y} \cdot \frac{dy}{dx} \). Here, \( \frac{1}{y} \) is the derivative of \( \ln y \) with respect to \( y \) and then multiplied by \( \frac{dy}{dx} \) since \( y \) is itself a function of \( x \). This step helps us elegantly handle situations where a variable like \( y \) depends on another variable \( x \).

Product Rule

In calculus, the product rule comes in handy when differentiating the products of two functions. The basic formula is: \( (uv)' = u'v + uv' \). This means that you derive the first function and multiply it by the second function, then add the first function multiplied by the derivative of the second function.
For the given problem, the product rule is applied to the expression \( e^y \sin x \). Here, the expressions \( e^y \) and \( \sin x \) are our two functions:

• Differentiate \( e^y \) with respect to \( x \), remembering that it involves \( y \), which is a function of \( x \). As a result, the derivative is \( e^y \cdot \frac{dy}{dx} \).
• The derivative of \( \sin x \) is \( \cos x \).

Applying the product rule gives us:

• \( \frac{d}{dx}[e^y \sin x] = e^y \cdot \cos x + e^y \sin x \cdot \frac{dy}{dx} \).

These expressions combine to express the derivative of the product neatly.

Derivative Calculation

Once you've applied the chain and product rules, it's time for the derivative calculation, which involves solving for \( \frac{dy}{dx} \). After differentiating both sides of the equation \( \ln y = e^y \sin x \), we get:

• \( \frac{1}{y} \cdot \frac{dy}{dx} = e^y \sin x \cdot \frac{dy}{dx} + e^y \cos x \).

The next step involves arranging all terms containing \( \frac{dy}{dx} \) on one side of the equation:

• This is rearranged to \( \frac{dy}{dx} \left( \frac{1}{y} - e^y \sin x \right) = e^y \cos x \).

Finally, solve for \( \frac{dy}{dx} \):

• We factor out \( \frac{dy}{dx} \) and divide by the remaining term: \( \frac{dy}{dx} = \frac{e^y \cos x}{\frac{1}{y} - e^y \sin x} \).

Simplifying the solution by multiplying the numerator and denominator by \( y \):

• Results in \( \frac{dy}{dx} = \frac{y e^y \cos x}{1 - y e^y \sin x} \).

This final form represents the derivative of \( y \) with respect to \( x \), providing a complete understanding of how the changes in \( x \) influence \( y \).