Problem 29 - Calculus Early Transcendentals

Question

\(25-30\) Find \(f^{\prime}(a)\) $$f(x)=\frac{1}{\sqrt{x+2}}$$

Step-by-Step Solution

Answer

\(f^{\prime}(a) = -\frac{1}{2(a+2)^{3/2}}\)

Step 1: Identify the function and the derivative rule needed

We are given the function \(f(x) = \frac{1}{\sqrt{x+2}}\). This function can be rewritten as \(f(x) = (x+2)^{-1/2}\). We need to find the derivative \(f^{\prime}(x)\). To do this, we will use the power rule for differentiation, which is \(\frac{d}{dx}[x^n] = n \cdot x^{n-1}\).

Step 2: Apply the power rule to differentiate

Using the power rule for derivatives on \(f(x) = (x+2)^{-1/2}\), we find:\[ f^{\prime}(x) = -\frac{1}{2} (x+2)^{-1/2-1} \cdot (1) \]The \(\cdot (1)\) is from the chain rule, as the derivative of \((x+2)\) with respect to \(x\) is 1.

Step 3: Simplify the derivative

Simplify \(f^{\prime}(x) = -\frac{1}{2} (x+2)^{-3/2}\). The negative exponent \(-3/2\) indicates the expression should be in the denominator in root form:\[ f^{\prime}(x) = -\frac{1}{2} \cdot \frac{1}{(x+2)^{3/2}} = -\frac{1}{2(x+2)^{3/2}} \]

Step 4: Evaluate the derivative at \(x=a\)

To find \(f^{\prime}(a)\), substitute \(x = a\) into the derivative:\[ f^{\prime}(a) = -\frac{1}{2(a+2)^{3/2}} \]This gives us the rate of change of the function \(f(x)\) at \(x = a\).

Key Concepts

Understanding the Power Rule

The power rule is a fundamental concept in calculus and a go-to tool for differentiating functions with exponents. It states that if you have a function of the form \(x^n\), then its derivative is \(n \cdot x^{n-1}\). This rule simplifies the process of taking derivatives by providing a straightforward formula. Instead of recalculating for each function, you apply this rule directly to find the rate of change. For example, if \(f(x) = x^3\), then the derivative, \(f^{\prime}(x)\), is \(3x^2\). This simple transformation is powerful because it works consistently across polynomial expressions, saving time and effort. When dealing with more complex functions that fit this form, the power rule becomes a trusty companion. Always remember to reduce the power by 1 and multiply by the original power—it's that easy!

Introducing the Chain Rule

The chain rule is essential for differentiating compositions of functions. It helps when you have a function within another function, such as \(f(g(x))\). Imagine you have an outer function and an inner function, both interacting to form a composite. The rule states: \( (f(g(x)))^{\prime} = f^{\prime}(g(x)) \cdot g^{\prime}(x) \). First, differentiate the outer function while keeping the inner function intact, and then multiply by the derivative of the inner function.

• Step 1: Differentiate the outer function \(f\).
• Step 2: Multiply by the derivative of the inner function \(g\).
• Step 3: Combine these elements for the full derivative.

In our example, the expression \((x+2)^{-1/2}\) involved using the chain rule. We took the derivative of the content of the square root and multiplied it by the derivative of the whole expression. Mastering this rule allows you to tackle complex layered functions with confidence.

Exploring Differentiation

Differentiation is a core concept in calculus that involves finding the derivative of a function. This process determines how a function changes over time, helping to understand the function's behavior. A derivative gives you the slope of the tangent line to the function's curve at any point, illustrating how the value of the function increases or decreases. The steps of differentiation require selecting the appropriate rules based on the function's form:

• Use the power rule for polynomial terms.
• Employ the chain rule for composite functions.
• Consider other rules for trigonometric, exponential, or logarithm functions.

With differentiation, you can analyze real-world problems involving rates, like velocity and acceleration. It equips you to manage varying conditions and reactions, making it indispensable in various scientific and engineering fields.

Essence of Function Simplification

Function simplification is key to making derivatives easier to handle. Before differentiating, simplify your function whenever possible. This step can reduce complexity and prevent errors, especially when dealing with fractions, roots, or negative exponents. In the given task, rewriting \(\frac{1}{\sqrt{x+2}}\) as \((x+2)^{-1/2}\) made it easier to apply the power rule directly. Simplification can involve:

• Converting roots to exponents.
• Reducing fractions to a simpler form.
• Expanding and rearranging terms.

By making functions more manageable, you can focus on applying derivative rules correctly, which enhances accuracy in problems. Simplification isn't just about making things smaller—it's about making them clearer and more workable.