Problem 5 - Calculus Early Transcendentals

Question

What is the derivative of \(y=e^{k x} ?\) For what values of \(k\) does this rule apply?

Step-by-Step Solution

Answer

Answer: The derivative of the function \(y = e^{kx}\) with respect to \(x\) is \(\frac{dy}{dx} = ke^{kx}\). This rule applies for any real value of \(k\).

Step 1: Recall the chain rule for differentiation

The chain rule for differentiation states that if a function \(y\) can be represented as a composition of two functions \(u\) and \(v\), i.e., \(y=u(v(x))\), then the derivative of \(y\) with respect to \(x\) is given by: $$\frac{dy}{dx} = \frac{du}{dv} \cdot \frac{dv}{dx}$$

Step 2: Identify the composition of functions

In this case, we can represent the given function as a composition of two functions by assigning \(u=e^v\) and \(v=kx\). This means that \(y=u(v(x))=e^{kx}\).

Step 3: Find the derivative of each function

Now, we will find the derivatives of each function with respect to their respective variables: For \(u=e^v\), the derivative is: $$\frac{du}{dv} = e^v$$ For \(v=kx\), the derivative is: $$\frac{dv}{dx} = k$$

Step 4: Apply the chain rule

Using the chain rule for the given function \(y=e^{kx}\), its derivative is: $$\frac{dy}{dx} = \frac{du}{dv} \cdot \frac{dv}{dx} = e^v \cdot k = e^{kx} \cdot k$$ So, the derivative of \(y=e^{kx}\) is: $$\frac{dy}{dx} = ke^{kx}$$

Step 5: Discuss the values of \(k\) for which the rule applies

The chain rule for differentiation applies to any real value of \(k\). The derivative obtained, \(ke^{kx}\), is valid for any real value of \(k\).

Key Concepts

Derivative

The concept of a derivative is central to calculus and describes how a function changes as its input changes. When we talk about the derivative of a function, we're looking at the rate of change or the slope of the function at any given point. In mathematical terms, the derivative of a function \( f(x) \) is denoted by \( f'(x) \) or \( \frac{df}{dx} \). It tells us how much \( y \) (the output) changes with a small change in \( x \) (the input). For exponential functions like \( y = e^{kx} \), the derivative measures the change in \( y \) as \( x \) changes. Understanding derivatives helps in many fields, including physics for understanding motion, in economics for understanding how different factors affect growth, and in data science for optimizing algorithms.

Exponential Functions

Exponential functions are a fundamental class of functions characterized by the independent variable appearing in the exponent. A classic example of an exponential function is \( y = e^{kx} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.718. This base gives rise to natural exponential functions which have useful properties in calculus and real-world applications.

• Rapid Growth or Decay: The function \( y = e^{kx} \) grows or decays exponentially depending on the sign of \( k \).
• Applications: Exponential functions model population growth, radioactive decay, interest calculations, and other processes where change occurs at a continuous rate.

When dealing with exponential functions, it's vital to recognize their unique growth patterns and how the parameter \( k \) influences their behavior, further shedding light on how specialization or focus on certain types of problems can positively impact understanding.

Differentiation

Differentiation is the process of finding a derivative, meaning determining the rate at which something changes. This process is core to calculus and is applied in diverse fields ranging from engineering to medicine.To differentiate a function like \( y = e^{kx} \), we employ rules such as the chain rule. The chain rule helps us tackle more complex functions by breaking them down into simpler parts. In this context, differentiation involves:

• Applying the Chain Rule: We identify the composite nature of \( y = e^{kx} \) with \( u = e^v \) and \( v = kx \), and apply the rule \( \frac{du}{dx} = \frac{du}{dv} \times \frac{dv}{dx} \).
• Executing Derivative Calculations: Calculating \( \frac{du}{dv} = e^v \) and \( \frac{dv}{dx} = k \), then combining via the chain rule to get \( \frac{dy}{dx} = ke^{kx} \).

Differentiation not only gives us the formula we seek but also strengthens our overall understanding of how functions behave and change, preparing us for more complex analysis and problem-solving in future topics.