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Chapter 2: Problem 88

Evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result. $$ f(x)=\frac{1}{\sqrt{x+4}}, \quad\left(0, \frac{1}{2}\right) $$

Short Answer

Expert verified
The second derivative of the function at the point (0,1/2) is \(\frac{3}{32}\).

Step by step solution

01

Function Interpretation

Consider the given function \(f(x)=\frac{1}{\sqrt{x+4}}\). This function can also be written as \(f(x)=(x+4)^{-1/2}\). With the power rule, this should simplify the differentiation process.
02

First Derivative

To find the first derivative, the power rule is used, \(\frac{d}{dx}(x^n) = n*x^{n-1}\). So, \(f'(x)=\frac{-1}{2}(x+4)^{-3/2}\). Now, it can be simplified by bringing the negative at the start and making the term in the bracket as the denominator under the square root, which yields \(f'(x)=-\frac{1}{2\sqrt{(x+4)^3}}\) as the first derivative.
03

Second Derivative

Taking derivative one more time for the second derivative. Apply the power rule again, \(f''(x)=\frac{3}{4}(x+4)^{-5/2}\). After simplification, the second derivative becomes \(f''(x)=\frac{3}{4\sqrt{(x+4)^5}}\).
04

Evaluate the Second Derivative at the Given Point

Substitute \(x=0\) in \(f''(x)=\frac{3}{4\sqrt{(x+4)^5}}\) and evaluating this gives \(f''(0)=\frac{3}{32}\). This is the value of the second derivative at point \(x=0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule Differentiation
Understanding power rule differentiation is like having a key to unlock one of calculus's most fundamental tools. It's a simple and efficient method for finding the derivatives of functions that are monomials鈥攅xpressions like ax^n where n is any real number.

The rule itself is straightforward: if you're given a function in the form of f(x) = x^n, then the derivative, f'(x), is given by n*x^(n-1). In essence, you take the power, multiply the function by it, and then decrease the power by one. Think of it as

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