Problem 38 - College Algebra and Calculus: An Applied Approach

Question

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=\frac{x+1}{\sqrt{x}} $$

Step-by-Step Solution

Answer

The derivative of the given function is \( f'(x) = 1 - \frac{1}{2\sqrt{x}} - \frac{1}{2x\sqrt{x}} \). The process involved using the chain rule, power rule, and quotient rule for differentiation.

Step 1: Rewrite the Function

First, rewrite the function in a form more suitable for differentiation. Divide each term in the numerator by the denominator to transform the original function into two separate terms. This yields \( f(x)=x\sqrt{x}^{-1}+ \sqrt{x}^{-1} \).

Step 2: Apply Power Rule and Chain Rule

Next, apply the power rule and chain rule for differentiation, and compute the derivatives of the two terms separately. The derivative of \( x\sqrt{x}^{-1} \) is 1-0.5(\( x^{-1} \)). The derivative of \( \sqrt{x}^{-1} \) is -0.5(\( x^{-1.5} \)).

Step 3: Simplify the Result

Once the derivatives have been calculated, combine and simplify them to yield the final derivative function. So, the derivative \(\ f'(x) = 1 - \frac{1}{2\sqrt{x}} - \frac{1}{2x\sqrt{x}} \).

Key Concepts

Power Rule

The Power Rule is a fundamental concept in differentiation, especially useful when dealing with polynomials or functions expressed in terms of powers of a variable. It is a handy tool because it provides a straightforward formula to find the derivative of such functions.

The rule states: If you have a function of the form \( f(x) = x^n \), where \( n \) is any real number, then the derivative of \( f(x) \) with respect to \( x \) is \( f'(x) = nx^{n-1} \). This formula allows you to quickly find the derivative of a function by simply multiplying the power of \( x \) by the coefficient and reducing the power by one.

In our exercise, we applied the Power Rule to both parts of the rewritten function, \( x\sqrt{x}^{-1} \) and \( \sqrt{x}^{-1} \). Keep in mind that \( \sqrt{x} \) can be rewritten as \( x^{0.5} \), and therefore \( \sqrt{x}^{-1} \) becomes \( x^{-0.5} \). Utilizing the Power Rule allows us to find the derivatives of these terms easily.

Chain Rule

The Chain Rule is an essential differentiation tool, particularly when dealing with composite functions. It allows you to take the derivative of a compound function, where one function is nested inside another.

Here's the basic idea: if you have a function \( f(g(x)) \), its derivative is given by \( f'(g(x)) \cdot g'(x) \). Essentially, you're taking the derivative of the outer function with respect to the inner function and then multiplying it by the derivative of the inner function itself.

In our exercise, the Chain Rule is subtly at play when dealing with \( x\sqrt{x}^{-1} \) and \( \sqrt{x}^{-1} \), because these terms can be seen as compositions of simpler functions such as powers and roots of \( x \). Applying the Chain Rule ensures that we correctly account for the changes in the inside functions as well, even though the outer dependencies seem simple.

Derivative

The derivative itself is a measure of how a function changes as its input changes. In simpler terms, it gives the rate of change or the slope of the function at any given point.

When we differentiate a function, we are essentially finding a new function, known as the derivative, which details the rates of change for all points of the original function. Recognizing the derivative is pivotal for solving problems involving rates of change, like velocity, acceleration, and slopes of curves.

• For example, if \( f(x) \) represents the position of an object at time \( x \), then \( f'(x) \) would represent its velocity.
• Similarly, if \( f(x) \) is a cost function, \( f'(x) \) could represent the marginal cost.

In our specific example, finding the derivative involved using the Power and Chain Rules to break down the function \( f(x)=\frac{x+1}{\sqrt{x}} \) into simpler parts. Then, computing the derivative of each part allowed us to understand how the entire function changes as \( x \) changes. The final derivative, \( f'(x) = 1 - \frac{1}{2\sqrt{x}} - \frac{1}{2x\sqrt{x}} \), offers valuable insights into the original function's behavior.