Problem 45 - Brief Applied Calculus

Question

Use the Generalized Power Rule to find the derivative of each function. $$ f(x)=\sqrt{1+\sqrt{x}} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{1}{4}(1+x^{1/2})^{-1/2}x^{-1/2} \).

Step 1: Rewrite the Function

First, express the function in a form suitable for differentiation. The given function is \[ f(x) = \sqrt{1+\sqrt{x}} \]which can be rewritten using exponents as:\[ f(x) = (1+x^{1/2})^{1/2} \].

Step 2: Apply the Generalized Power Rule

Apply the Generalized Power Rule, which states that the derivative of \((g(x))^n\) is \[n(g(x))^{n-1}\cdot g'(x)\]. Here, \(n = \frac{1}{2}\) and \(g(x) = 1+x^{1/2}\), so we start by differentiating using the rule:\[ \frac{1}{2}(1+x^{1/2})^{-1/2} \cdot (1 + x^{1/2})' \].

Step 3: Find the Inner Function's Derivative

Find the derivative of the inner function, \( g(x) = 1 + x^{1/2} \). The derivative of \(1\) is \(0\), and the derivative of \(x^{1/2}\) is \(\frac{1}{2}x^{-1/2}\).Thus, \(g'(x) = \frac{1}{2}x^{-1/2}\).

Step 4: Combine the Results

Combine the results from Steps 2 and 3:\[ f'(x) = \frac{1}{2}(1+x^{1/2})^{-1/2}\cdot\frac{1}{2}x^{-1/2} \].Multiply the constants and simplify the expression:\[ f'(x) = \frac{1}{4}(1+x^{1/2})^{-1/2}x^{-1/2} \].

Key Concepts

Derivative Calculation

Derivative calculation is a fundamental concept in calculus. It describes the process of finding the rate at which a function's value changes as its input changes. The derivative provides a mathematical expression for the slope of the tangent line at any point on the function's curve.
To calculate the derivative of the function \( f(x) = \sqrt{1+\sqrt{x}} \), we start by rewriting the original expression using exponents. This involves expressing the square root terms in a form that is more amenable to the rules of differentiation. In this exercise, we write \( f(x) \) as \( (1+x^{1/2})^{1/2} \).
Recognizing that the function is in this form allows us to apply the Generalized Power Rule effectively.

Function Differentiation

Function differentiation involves finding the rate of change of a function with respect to its variable. This usually requires applying specific rules that simplify the differentiation process.
In the given exercise, we use the Generalized Power Rule for differentiation. This rule is helpful for functions that can be expressed in the form \((g(x))^n\). The Generalized Power Rule tells us that the derivative \( (g(x))^n \) is given by \( n(g(x))^{n-1} \cdot g'(x) \).

• Identify: Firstly, identify the values of \(n\) and \(g(x)\). Here, \(n = \frac{1}{2}\) and \(g(x) = 1 + x^{1/2}\).
• Differentiate: Calculate the derivative of the inner function \(g(x)\), which involves using basic power rule derivatives. In this case, the inner derivative \(g'(x)\) finds \(x^{1/2}\) is \(\frac{1}{2}x^{-1/2}\).

Once \(g'(x)\) is known, you incorporate it into the formula from the generalized power rule.

Calculus Problem Solving

Solving calculus problems like this one requires a systematic approach, understanding of rules, and manipulation of expressions.
Let's break down the process:

• Rewriting Functions: It’s essential to express functions using exponents, making differentiation straightforward.
• Rule Application: Understand and apply differentiation rules like the Generalized Power Rule.
• Simplifying Results: Combine and simplify terms after applying derivatives to achieve the final form.

This problem involves each of these steps clearly. After deriving the required expressions through rules and simplifications, you achieve the simplified derivative: \( f'(x) = \frac{1}{4}(1+x^{1/2})^{-1/2}x^{-1/2} \). Practice and familiarity with these methods are key in mastering calculus problem solving.