Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the indicated derivative. \(D_{x} e^{\sqrt{x+2}}\)

Step-by-Step Solution

Answer

The derivative is \( \frac{e^{\sqrt{x+2}}}{2\sqrt{x+2}} \).

Step 1: Identify the Function

We start with the function given as: \( f(x) = e^{\sqrt{x+2}} \). Our goal is to differentiate this function with respect to \( x \).

Step 2: Apply Chain Rule

The function is a composition of two functions: the outer function is \( g(u) = e^u \) and the inner function is \( u(x) = \sqrt{x+2} \). The chain rule tells us to take the derivative of the outer function with respect to the inner function, then multiply by the derivative of the inner function with respect to \( x \).

Step 3: Differentiate the Outer Function

Differentiate the outer function \( g(u) = e^u \) with respect to \( u \). We have \( \frac{d}{du} e^u = e^u \). Substitute \( u = \sqrt{x+2} \) back into the result to get \( e^{\sqrt{x+2}} \).

Step 4: Differentiate the Inner Function

Differentiate the inner function \( u(x) = \sqrt{x+2} \) with respect to \( x \). Using the chain rule and derivative of square root function, we find \( \frac{d}{dx} \sqrt{x+2} = \frac{1}{2\sqrt{x+2}} \).

Step 5: Combine Results

Multiply the results of the derivatives from Step 3 and Step 4: \( e^{\sqrt{x+2}} \times \frac{1}{2\sqrt{x+2}} \).

Step 6: Simplify the Expression

The simplified expression becomes: \( \frac{e^{\sqrt{x+2}}}{2\sqrt{x+2}} \).

Key Concepts

Chain Rule

The chain rule is a fundamental derivative rule in calculus for finding the derivative of a composition of functions. It is particularly powerful when dealing with situations where one function is nested inside another. To use the chain rule effectively:

• Identify the inner and outer functions in the composition.
• Differentiating the outer function with respect to the inner one.
• Multiply the derivative by the derivative of the inner function.

For example, consider differentiating a function like \(f(x) = e^{\sqrt{x+2}}\). Here, the outer function is \(e^u\) and the inner is \(\sqrt{x+2}\). By applying the chain rule, first differentiate the outer function with respect to \(u\), then multiply by the derivative of the inner function with respect to \(x\).
This approach allows us to tackle complex differentiation tasks by breaking them down into more manageable steps.

Function Differentiation

Function differentiation is the process of finding the derivative of a function, which tells us how the function changes at any given point. When differentiating, it's essential to identify the type of function you are working with, whether it's a polynomial, exponential, or involves a square root.

• A polynomial's derivative follows basic power rules.
• For exponential functions, you often find that the derivative involves the original function itself.
• Square root functions require careful application of chain rule or power rule.

Differentiation is crucial in calculus as it provides rates of change, which are applicable in real-world scenarios such as physics for velocity and acceleration.

Exponential Functions

Exponential functions have the form \(e^x\), where \(e\) is Euler's number, approximately 2.718. These functions grow rapidly and their behavior is unique. Differentiating exponential functions is straightforward because:

• The derivative of \(e^x\) is simply \(e^x\).
• This property applies to functions of the form \(e^{u(x)}\), with the chain rule needed when \(u(x)\) is not just \(x\).

For example, to differentiate \(e^{\sqrt{x+2}}\), identify the exponent \(u\) as \(\sqrt{x+2}\) and apply the chain rule. This simplicity makes exponential functions a favorite in modeling real-world phenomena where growth or decay rates are involved.

Square Root Functions

Square root functions, such as \(\sqrt{x}\), require careful differentiation due to their form. The derivative of a square root function is not as intuitive as linear functions. To differentiate a square root function:

• Express the square root as a power: \(\sqrt{x} = x^{1/2}\).
• Use the power rule: \(\frac{d}{dx}(x^{n}) = nx^{n-1}\).

Thus, the derivative of \(\sqrt{x}\) becomes \(\frac{1}{2\sqrt{x}}\). For functions like \(\sqrt{x+2}\), apply the chain rule to differentiate inside the square root first, resulting in a more complex form. Adjustments like these are key in solving derivatives involving square roots correctly.