Question

Graphical Reasoning In Exercises 49 and 50 , use a graphing utility to graph the function and its derivative in the same viewing window. Label the graphs and describe the relationship between them. \(f(x)=\frac{1}{\sqrt{x}}\)

Short answer

The derivative of the function \(f(x) = \frac{1}{\sqrt{x}}\) is \(f'(x) = -\frac{1}{2x\sqrt{x}}\). The graphs of the function and its derivative confirm this by showing that the function decreases as x increases, reflecting the negative values of the derivative.

Step-by-step solution

Function Analysis

The given function is \(f(x) = \frac{1}{\sqrt{x}}\). Examine this function and understand its characteristics, particularly that it exists for all \(x > 0\).

Find the Derivative

The derivative of the function \(f(x) = \frac{1}{\sqrt{x}}\) can be found using the power rule. This yields \(f'(x) = -\frac{1}{2x\sqrt{x}}\).

Graph Initial Function

Graph the function \(f(x) = \frac{1}{\sqrt{x}}\) using a graphing utility. This graph should display a curve that decreases as x increases, beginning from a high point at \(x > 0\) and approaching zero as \(x\) tends towards infinity.

Graph the Derivative

Now graph the derivative \(f'(x) = -\frac{1}{2x\sqrt{x}}\) obtained in the same viewing window. This curve should show negative values for all \(x > 0\) which means the initial function \(f(x)\) is decreasing for all \(x > 0\).

Analyze the Relationship

The relationship between the graphs of the function and its derivative is that the function is decreasing wherever the derivative is negative. This shows that the derivative of a function describes the rate of change of the function.