Problem 41 - Precalculus with Limits

Question

In Exercises 29-42, find the derivative of the function. \(f(x) = \dfrac{1}{\sqrt{x-9}}\)

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = \dfrac{1}{\sqrt{x-9}}\) is \(f'(x) = -\dfrac{1}{2\sqrt{(x-9)^3}}\).

Step 1: Rewrite the function

First, rewrite the function in a form that's suitable for using the power rule. We write \(f(x) = \dfrac{1}{\sqrt{x-9}}\) as \(f(x) = (x-9)^{-\frac{1}{2}}\).

Step 2: Apply the power rule

Then, apply the power rule, which states that the derivative of \(x^n\) is \(nx^{n-1}\). Following this rule, the derivative of \((x-9)^{-\frac{1}{2}}\) is \(-\frac{1}{2}(x-9)^{-\frac{3}{2}}\).

Step 3: Apply the chain rule

Finally, apply the chain rule which takes the derivative of the outside function multiplied by the derivative of the inside function. The derivative of \(x-9\) is 1. So our final derivative is \(-\frac{1}{2}(x-9)^{-\frac{3}{2}} * 1\).

Step 4: Simplify the derivative

Simplifying the derivative gives us \(f'(x) = -\dfrac{1}{2}(x-9)^{-\frac{3}{2}}\), which can be rewritten as \(f'(x) = -\dfrac{1}{2\sqrt{(x-9)^3}}\).

Key Concepts

Power Rule

The power rule is a fundamental technique in calculus employed to find the derivative of expressions given in the form of a power. When you have a function expressed as

• \(f(x) = x^n\)

,it can be easily differentiated using this rule. Simply multiply the function by the power and then decrease the original power by one. This gives you the derivative \(f'(x) = nx^{n-1}\).

Applying the power rule simplifies many derivative computations as it turns potentially complex expressions into manageable forms. For instance, if you rewrite \(\frac{1}{\sqrt{x-9}}\) into \((x-9)^{-\frac{1}{2}}\), the power rule makes it straightforward to differentiate by plugging \(-\frac{1}{2}\) into our formula. This foundational concept is often one of the first rules introduced in a calculus course, providing a simple technique for tackling derivatives of polynomial-like functions with ease.

Chain Rule

The chain rule is essential when dealing with composite functions, which are functions formed by the combination of two or more functions. If you have a composition of functions such as \(g(f(x))\), the chain rule is used to find its derivative. It works by taking the derivative of the outer function and multiplying it by the derivative of the inner function. This can be thought of as "chaining" the derivatives together.

For the function \((x-9)^{-\frac{1}{2}}\), we recognize that it's composed of the inner function \(x-9\) and the outer function \(u^{-\frac{1}{2}}\) (calling \(u = x-9\)). The chain rule dictates that we first differentiate the outer function, resulting in \(-\frac{1}{2}(u)^{-\frac{3}{2}}\), and then multiply it by the derivative of the inner function, \(1\). Therefore, the composite derivative simplifies to \(-\frac{1}{2}(x-9)^{-\frac{3}{2}}\), showcasing the elegant power of the chain rule in calculus.

Derivative of a Function

Calculating the derivative of a function unveils how the function changes at any given point. It is essentially the slope of the tangent line to the function's graph at a specific point. The derivative provides not only the rate of change but also highlights where a function increases, decreases, or remains constant.

When finding derivatives, we often employ several established rules, like the power rule and the chain rule, to systematically break down complex expressions. For the function \(f(x) = \frac{1}{\sqrt{x-9}}\), determining the derivative involves rewriting the function to a differentiable form as \((x-9)^{-\frac{1}{2}}\), applying the power rule to find the derivative of the overall function, and finally utilizing the chain rule to handle its composite nature. The derivative \(f'(x) = -\frac{1}{2\sqrt{(x-9)^3}}\) tells us precisely how \(f(x)\) behaves, offering insight into its graphical slope and tendency that can be utilized in broader analyses or applications.