Problem 25 - Thomas Calculus in SI Units

Question

Find \(d y / d x.\) $$\ln y=e^{y} \sin x$$

Step-by-Step Solution

Answer

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

Step 1: Differentiate Both Sides with Respect to x

Start by taking the derivative of the equation \(\ln y = e^{y} \sin x\) with respect to \(x\). Using implicit differentiation for \(\ln y\) gives \(\frac{1}{y} \cdot \frac{dy}{dx}\). The derivative of the right side requires the product rule.

Step 2: Apply Product Rule to the Right Side

The derivative of \(e^{y} \sin x\) using the product rule is \(\frac{d}{dx}(e^{y} \cdot \sin x) = e^{y} \cdot \frac{d}{dx}(\sin x) + \sin x \cdot \frac{d}{dx}(e^{y})\). Since \(\frac{d}{dx}(\sin x) = \cos x\) and \(\frac{d}{dx}(e^{y}) = e^{y} \cdot \frac{dy}{dx}\), the right side becomes \(e^{y} \cos x + \sin x \cdot e^{y} \frac{dy}{dx}\).

Step 3: Rearrange the Equation

Set the derivatives equal: \(\frac{1}{y} \cdot \frac{dy}{dx} = e^{y} \cos x + \sin x \cdot e^{y} \frac{dy}{dx}\). Move all terms involving \(\frac{dy}{dx}\) to one side of the equation: \(\frac{1}{y} \cdot \frac{dy}{dx} - \sin x \cdot e^{y} \frac{dy}{dx} = e^{y} \cos x\).

Step 4: Factor out \(\frac{dy}{dx}\)

Factor \(\frac{dy}{dx}\) from the left side to get: \(\frac{dy}{dx} \left( \frac{1}{y} - \sin x \cdot e^{y} \right) = e^{y} \cos x\).

Step 5: Solve for \(\frac{dy}{dx}\)

Solve for \(\frac{dy}{dx}\) by dividing both sides by \(\left( \frac{1}{y} - \sin x \cdot e^{y} \right)\): \[ \frac{dy}{dx} = \frac{e^{y} \cos x}{\frac{1}{y} - \sin x \cdot e^{y}} \].

Key Concepts

Product Rule in Differentiation

When working with derivatives, especially in more complex functions, the product rule is a handy tool. If you're given a function that is the product of two other functions, the product rule will help you differentiate it correctly. Let's say you have a function that looks like this:

• \( u(x) \cdot v(x) \)
The product rule says that the derivative of this function would be:
• \( u'(x) \cdot v(x) + u(x) \cdot v'(x) \)
In simpler terms, it tells you to differentiate each part separately and then add them up together in a specific way. In our exercise, the function \( e^{y} \sin x \) is composed of both an exponential component and a trigonometric one. This complexity makes the product rule essential for finding its derivative properly. Each part is differentiated separately: \( e^{y} \) is multiplied by the derivative of \( \sin x \), and \( \sin x \) is multiplied by the derivative of \( e^{y} \). Once calculated, simply add them together for the full derivative.

Understanding Derivatives

Derivatives are fundamental in understanding how functions change. They represent the rate of change of a function's output in relation to a change in input. Think of it as measuring how quickly something is changing at any given point. This concept is key in both pure and applied calculus. In this problem, derivatives help us find \( \frac{dy}{dx} \), which represents the rate of change of \( y \) with respect to \( x \).
To differentiate implicitly, like we do with \( \ln y = e^{y} \sin x \), we assume \( y \) is a function of \( x \) and differentiate both sides of the equation with respect to \( x \). This method is particularly useful when \( y \) is not explicitly defined as a function of \( x \). We use our knowledge of basic rules of differentiation, such as the chain rule, product rule, and the derivatives of simpler functions like polynomials, exponentials, and trigonometric functions, to tackle more complex equations. Understanding these differentiation rules is crucial to solving calculus problems effectively.

Approaching Calculus Problem Solving

When approaching calculus problems, a structured strategy makes tackling even the toughest problems much easier. To solve the given problem, follow these general steps:

• Start by understanding the problem. Identify the type of function and relationship you are dealing with. Here, notice that both sides of the equation \( \ln y = e^{y} \sin x \) involve different types of functions - logarithms, exponentials, and trigonometry.


• Apply the right calculus tools. Use implicit differentiation by taking the derivative of each side with respect to \( x \). Be ready to apply the product rule when needed, especially when dealing with a product of functions like \( e^{y} \sin x \).


• Carefully rearrange and solve the resulting derivative equation. Move all the terms involving \( \frac{dy}{dx} \) to one side of the equation, then factor them out.


• In the final step, isolate \( \frac{dy}{dx} \) by dividing both sides of the equation appropriately, ensuring you arrive at the correct expression for the derivative.

By methodically following these steps and ensuring a firm understanding of the fundamental principles of calculus, you'll be well-equipped to solve complex problems efficiently.