Problem 31 - University Calculus: Early Transcendentals

Question

Find \(d y\). $$y=e^{\sqrt{x}}$$

Step-by-Step Solution

Answer

\( dy = \frac{e^{\sqrt{x}}}{2\sqrt{x}} dx \).

Step 1: Identify the Function to Differentiate

We begin by identifying the function given in the problem, which is \( y = e^{\sqrt{x}} \). Our task is to find \( dy \), the differential of this function.

Step 2: Determine the Form of the Chain Rule

The function \( y = e^{\sqrt{x}} \) is a composition of two functions: the exponential function and the square root function. We will apply the chain rule which states \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \). In this case, \( f(u) = e^u \) and \( g(x) = \sqrt{x} \).

Step 3: Differentiate the Outer Function

Differentiate the outer function, \( f(u) = e^u \), with respect to \( u \). The derivative of \( e^u \) is \( e^u \). Therefore, \( f'(g(x)) = e^{\sqrt{x}} \).

Step 4: Differentiate the Inner Function

Differentiate the inner function, \( g(x) = \sqrt{x} \), with respect to \( x \). Recall that \( \sqrt{x} = x^{1/2} \). Differentiating gives \( g'(x) = \frac{1}{2}x^{-1/2} \) or \( \frac{1}{2\sqrt{x}} \).

Step 5: Apply the Chain Rule

Apply the chain rule: multiply the derivative of the outer function by the derivative of the inner function. Thus, the derivative \( \frac{dy}{dx} = e^{\sqrt{x}} \cdot \frac{1}{2\sqrt{x}} \).

Step 6: Express the Differential dy

The differential \( dy \) is given by \( dy = \frac{dy}{dx} \cdot dx \). Substituting the expression for \( \frac{dy}{dx} \), we get \( dy = \frac{e^{\sqrt{x}}}{2\sqrt{x}} dx \).

Key Concepts

Chain Rule

The chain rule is a fundamental tool in differential calculus. It is used to differentiate a composition of functions. In simpler terms, when you have a function within another function, this rule helps you find the derivative of the whole composite function. The chain rule is represented mathematically as:

• \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \)

In the given exercise, the function to be differentiated is \( y = e^{\sqrt{x}} \). Here, we see a composite of two functions:

• The inner function \( g(x) = \sqrt{x} \)
• The outer function \( f(u) = e^u \)

Applying the chain rule involves two steps:

• First, differentiate the outer function \( e^u \) while keeping the inner part \( \sqrt{x} \) constant. This yields \( e^{\sqrt{x}} \).
• Next, differentiate the inner function \( \sqrt{x} \) to get \( \frac{1}{2\sqrt{x}} \).

The final derivative is the product of these components. The chain rule simplifies the process of differentiation in cases where functions are nested within each other.

Exponential Functions

Exponential functions are a crucial part of calculus and appear frequently in mathematical equations. These functions have the form \( f(x) = a^x \), where \( a \) is a constant, often Euler's number \( e \) (approximately 2.71828). A special feature of the exponential function involving base \( e \) is that its derivative is the function itself:

• \( \frac{d}{dx}[e^x] = e^x \)

In this exercise, we dealt with \( y = e^{\sqrt{x}} \). The exponential part, \( e^{\sqrt{x}} \), follows this same derivative rule since the base is \( e \).

The derivative of \( e^u \) with respect to \( u \) stays \( e^u \), making exponential differentiation straightforward. Despite its simplicity, the exponential function's characteristic of maintaining its structure post differentiation is highly valued in various calculus-based applications.

Differentiation Techniques

Differentiation is the process of finding the rate at which one quantity changes with respect to another. Various techniques simplify this process, with each technique having its special application area.

• The power rule: Used for functions of the form \( x^n \), with the derivative given by \( nx^{n-1} \).
• The product rule: Useful when differentiating products of two functions.
• The quotient rule: For differentiating a ratio of two functions.

In the context of our exercise, we used the chain rule, which is ideal for functions within functions. Moreover, understanding exponential function differentiation is crucial as well. By combining these methods, we efficiently solve problems like finding \( dy = \frac{e^{\sqrt{x}}}{2\sqrt{x}} dx \).

Differentiation techniques, when correctly applied, enable us to unpack complex expressions into simpler, solvable parts, easing the calculus journey for students and practitioners alike.