Problem 11 - Thomas Calculus

Question

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. \begin{equation}y=x e^{x}-e^{x}\end{equation}

Step-by-Step Solution

Answer

The derivative is \(\frac{dy}{dx} = xe^{x}\).

Step 1: Identify the Expression

The given expression is \(y = x e^{x} - e^{x}\). To find the derivative \(\frac{dy}{dx}\), we'll apply differentiation with respect to \(x\).

Step 2: Differentiate Each Term Separately

Identify the components: \(u = x\) and \(v = e^{x}\) for the term \(xe^{x}\). Use the product rule \( \frac{d}{dx}(uv) = u'v + uv' \).

Step 3: Apply the Product Rule

For the term \(xe^{x}\), differentiate to get \(1 \cdot e^{x} + x \cdot e^{x} = e^{x} + xe^{x}\).

Step 4: Differentiate the Second Term

The derivative of \(e^{x}\) with respect to \(x\) is \(e^{x}\).

Step 5: Combine the Results

Combine the derivatives: \(e^{x} + xe^{x} - e^{x}\). The \(e^{x}\) terms cancel, leaving us with \(xe^{x}\).

Step 6: State the Final Derivative

The derivative of \(y\) with respect to \(x\) is \(\frac{dy}{dx} = xe^{x}\).

Key Concepts

Understanding Derivatives

In calculus, a derivative represents the rate at which a function is changing at any given point. It's a fundamental concept that allows us to describe how a function evolves. When we take the derivative of a function, we are essentially identifying the function's instantaneous rate of change. For example, if you have a function describing position over time, its derivative would represent velocity. This is crucial in understanding how quantities vary and is widely used in fields such as physics, engineering, and economics.
The process of finding a derivative is called differentiation. It involves applying various rules and techniques to determine how one quantity changes in relation to another. The result is often denoted as \( \frac{dy}{dx} \), which reads as 'the derivative of \( y \) with respect to \( x \)'. Here, \( y \) is the function and \( x \) is the variable with respect to which we are differentiating.

The Product Rule in Differentiation

The product rule is a useful technique for finding the derivative of a product of two functions. When you have an expression where functions are multiplied together, like \( u \times v \), their derivative isn't just a simple matter of multiplying the derivatives. Instead, you need to apply the product rule.
The product rule formula is: \( \frac{d}{dx}(uv) = u'v + uv' \). This means to find the derivative of \( u \times v \):

• First, find the derivative of \( u \), denoted as \( u' \), and then multiply it by \( v \).
• Then, keep \( u \) as it is and multiply it by the derivative of \( v \), denoted as \( v' \).
• Add these two results together.

The product rule is particularly handy when dealing with complicated expressions, just like in the term \( xe^{x} \) from our original exercise. By breaking it down into simpler parts, we ensure that we correctly find how the combined function changes.

Differentiation in Practice

Differentiation is putting the concept of derivatives into practice by applying rules like the product rule, chain rule, and others to calculate derivatives of various functions. It involves recognizing the structure of the function and selecting the appropriate methods to differentiate it.
In our exercise, we had the function \( y = x e^{x} - e^{x} \). To find the derivative \( \frac{dy}{dx} \), we needed to:

• Identify components of the expression: one with a product \( x e^{x} \) and the other a simple term \( e^{x} \).
• Apply the product rule to \( x e^{x} \) to get \( e^{x} + xe^{x} \).
• Find the derivative of \( e^{x} \), which is \( e^{x} \).
• Combine the results from each term by performing subtraction, resulting in \( xe^{x} \).

This step-by-step approach illustrates how differentiation enables us to systematically break down expressions and understand their rates of change effectively.