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Understanding the derivative is fundamental when dealing with the slope of a tangent line. The derivative of a function gives us the rate at which the function's value changes as the input changes. Think of it like measuring the steepness of a hill—how quickly you're climbing or descending at any moment. The derivative is like a mathematical tool that helps us find this instantaneous rate of change.

In our problem, the function is given by the equation \( y = \frac{1}{\sqrt{x}} \), and we want to find the slope of the tangent at a specific point, \( x = 4 \). To achieve this, the very first step is to derive the equation, which transforms it into a form that tells us exactly how the function is behaving at every point along its curve. This derived equation is what empowers us to compute the slope of the tangent line at any given \( x \) value. So remember, the derivative is essentially a function's very own slope calculator!

The power rule is a straightforward yet powerful tool in calculus that simplifies the process of differentiation. Essentially, it allows you to find the derivative of a power function quickly. The rule states:

• If \( y = x^n \), then the derivative \( \frac{dy}{dx} = n \cdot x^{n-1} \).

In our exercise, the function is initially \( y = \frac{1}{\sqrt{x}} \), which isn’t directly a power, but can be rewritten as \( y = x^{-1/2} \) using properties of exponents. This transformation is crucial as it enables the application of the power rule.By applying the power rule here, you differentiate \( x^{-1/2} \) to get \( -\frac{1}{2} \cdot x^{-3/2} \). The power rule allows us to easily handle such transformations and swiftly find derivatives, making calculus more manageable. It takes the complexity of polynomial differentiation and gives you a handy shortcut.

Differentiation is a process in calculus where you find the derivative of a function. To differentiate successfully, following systematic steps is key.Step-by-step guide:

• Step 1: Understand the Problem - Identify which function you need to differentiate and where you need the slope of the tangent line.
• Step 2: Rewrite the Function - Convert the function into a form where differentiation is easier. In our case, \( y = \frac{1}{\sqrt{x}} \) becomes \( y = x^{-1/2} \).
• Step 3: Differentiate - Use the power rule to find the derivative of the rewritten function. This step gives us \( \frac{dy}{dx} = -\frac{1}{2} x^{-3/2} \).
• Step 4: Substitute the Point - Plug the specific \( x \) value into the derivative to find the exact slope at that point.
• Step 5: Calculate the Slope - Simplify the derivative's value upon substitution to find the slope of the tangent line. For \( x = 4 \), the slope calculated is \( -\frac{1}{16} \).

Each step is intentional, making differentiation a manageable task. It breaks down the problem and uses calculus rules to arrive at an accurate solution.