Problem 37 - Precalculus with Limits

Question

In Exercises 29-42, find the derivative of the function. \(f(x) = \sqrt{x-11}\)

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = \sqrt{x-11}\) is \(f'(x) = \frac{1}{2\sqrt{x-11}}\).

Step 1: Express the Root Function as a Power Function

We begin by rewriting the square root function as a power function, making it easier to differentiate. So, \(f(x) = \sqrt{x-11}\) can be expressed as \(f(x) = (x-11)^{0.5}\).

Step 2: Apply the Power Rule

We can now apply the 'power rule', which states that if \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\). In this case, we treat \(x-11\) as our \(x\) and \(0.5\) as our \(n\) and apply the chain rule. Applying these rules of differentiation, we get: \(f'(x) = \frac{0.5(x-11)^{-0.5}}{dx/dx} = 0.5(x-11)^{-0.5}\). Therefore \(f'(x) = 0.5(x-11)^{-0.5}\).

Step 3: Apply the Chain Rule

Now, the chain rule has to be applied together with the derivative of \(x-11\) since we treated \(x-11\) as our \(x\) in the previous step. The chain rule states that the derivative of a composite function is the derivative of the outside function times the derivative of the inside function. The derivative of \(x-11\) is simply 1. Therefore, applying the chain rule we multiply our result from step 2 by 1, the derivative of \(x - 11\), which gives us: \(f'(x) = 0.5(x-11)^{-0.5} \cdot 1\).

Step 4: Simplify the Result

The derivative can be simplified by rewriting \(0.5(x-11)^{-0.5}\) as \(\frac{1}{2\sqrt{x-11}}\). Hence, the derivative of the function \(f(x) = \sqrt{x-11}\) is \(f'(x) = \frac{1}{2\sqrt{x-11}}\).

Key Concepts

Power Rule for Derivatives

The power rule is a fundamental technique in calculus for taking the derivative of functions of the form \(f(x) = x^n\), where \(n\) is any real number. The rule states that the derivative of such a function is \(f'(x) = nx^{n-1}\). This rule makes it simple to differentiate functions where the variable \(x\) is raised to a power.

When applying the power rule to a square root function, you can first express the square root as an exponent of \(1/2\) since \(\sqrt{x} = x^{1/2}\). For example, for \(f(x) = \sqrt{x-11}\), you write it as \(f(x) = (x-11)^{0.5}\). You then differentiate it as you would with any power of \(x\), resulting in \(f'(x) = 0.5(x-11)^{-0.5}\).

Understanding the power rule is crucial because it's commonly used not just in simple derivatives but also in more advanced calculus concepts where variables are raised to any power, including fractional and negative exponents.

Chain Rule in Calculus

The chain rule is an essential tool in calculus for finding the derivative of composite functions. When a function is composed of one function inside another, the chain rule allows us to differentiate the entire function in a systematic way. It is expressed as \(\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)\), where \(f\) is the outside function and \(g\) is the inside function.

Using the example \(f(x) = \sqrt{x-11}\), or \(f(x) = (x-11)^{0.5}\), the outside function is \(u^{0.5}\) where \(u = x-11\), and the inside function is \(x-11\). Applying the chain rule means taking the derivative of the outside function \(0.5 u^{-0.5}\), using \(u\) as a placeholder, and then multiplying by the derivative of the inside function, which is 1 since the derivative of \(x-11\) is 1. Hence, the application of the chain rule simplifies the derivative to \(0.5(x-11)^{-0.5}\cdot 1\).

The chain rule enables students to handle more complex differentiations with ease, extending their ability to manage a multitude of functions that are compositions of simpler ones.

Simplifying Derivatives

Simplifying derivatives is the process of making the derivative of a function as straightforward and clear as possible, often by rewriting it in a more conventional form. This is particularly useful when the derivative results in an expression that is difficult to interpret or use in further calculations.

As we have seen with the function \(f(x) = (x-11)^{0.5}\), the derivative was initially found to be \(f'(x) = 0.5(x-11)^{-0.5}\). By simplifying derivatives, it can be further written in a more conventional form, \(f'(x) = \frac{1}{2\sqrt{x-11}}\), which is easier to read and comprehend.

Key Steps in Simplifying Derivatives:

• Recognize and rewrite expressions using exponents and roots to their simplest forms.
• Factor common terms and cancel out terms when appropriate.
• Rewrite complex fractions into simpler, more readable fractions when possible.

Understanding how to simplify a derivative not only makes it easier to work with but also improves one's ability to recognize patterns and solve calculus problems more rapidly.