Problem 19 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Let \(f(x)=x^{-\frac{1}{2}}\). Use the limit de nition of derivative to show that \(f^{\prime}(x)=-\frac{1}{2} x^{-\frac{3}{2}}\).

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = x^{-\frac{1}{2}}\) is \(f^{\prime}(x) = -\frac{1}{2} x^{-\frac{3}{2}}\).

Step 1: Write the function

First, note the function given, which is \(f(x) = x^{-\frac{1}{2}}\). This will be the function we will be differentiating.

Step 2: Apply the limit definition of the derivative

Next, apply the limit definition of the derivative. This is \(\lim_{{\Delta x} \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}\). Substituting \(f(x)\) we get \(\lim_{{\Delta x} \to 0} \frac{(x+\Delta x)^{-\frac{1}{2}} - x^{-\frac{1}{2}}}{\Delta x}\).

Step 3: Simplify the expression

The next step is to simplify the expression within the limit. This involves finding a common denominator for the terms in the numerator. Multiply the first term by \(\frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}}\) and the second term by \(\frac{(x+\Delta x)^{\frac{1}{2}}}{(x+\Delta x)^{\frac{1}{2}}}\). This leads to \(\lim_{{\Delta x} \to 0} \frac{ x^{\frac{1}{2}} - (x+\Delta x)^{\frac{1}{2}}}{\Delta x \cdot x^{\frac{1}{2}} \cdot (x + \Delta x)^{\frac{1}{2}}}\).

Step 4: Take the limit as \(\Delta x \) approaches zero

Finally, compute the limit as \(\Delta x\) goes to zero. This simplifies the denominator to \(x\cdot x^{\frac{1}{2}}\). Thus we get \(f^{\prime}(x) = -\frac{1}{2} x^{-\frac{3}{2}}\).

Key Concepts

Limit Definition

The concept of a derivative is foundational in calculus, tracing its roots back to the idea of a limit. In mathematics, the limit definition of a derivative provides a way to compute the slope of a tangent line to a function's graph at any given point. Starting with the function, we denote the derivative of a function \( f(x) \) at a point \( x \) as \( f'(x) \). This derivative is determined by the formula:

• \( f'(x) = \lim_{{\Delta x} \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} \)

This expression represents the slope of the secant line as the interval \( \Delta x \) becomes infinitesimally small, essentially becoming the slope of the tangent line.
When working through exercises like the one provided, applying the limit definition involves a few steps:

• Substitute the function into the limit formula.
• Simplify the expression within the limit.
• Evaluate the limit as \( \Delta x \to 0 \).

The challenge often lies in algebraically simplifying the expression to make the limit computable, as demonstrated in the exercise.

Differentiation

Differentiation is the process by which we find the derivative of a function. It is a major operation in calculus alongside integration. The derivative essentially measures how a function's output value changes as its input changes. In our example, the function \( f(x) = x^{-\frac{1}{2}} \) represents a power function. Differentiating this type of function involves reducing it down to a more recognizable form, which allows us to apply rules and techniques efficiently.
In practical application, differentiation is not only about finding numeric values. It offers insights into the behavior of functions

• How steep or flat they are at certain points.
• Where they increase or decrease.
• Where they reach maximum or minimum values.

By using the limit definition during differentiation, we obtain the derivative \( f'(x) = -\frac{1}{2} x^{-\frac{3}{2}} \), revealing how our original function changes with respect to \( x \). This confirms the correspondence between the function and its rate of change over its domain.

Power Rule

The power rule is one of the easiest rules in calculus used to differentiate power functions. Its straightforward application makes solving many derivative problems simpler and more intuitive. This rule states that if you have a function of the form \( f(x) = x^n \), where \( n \) is any real number, the derivative with respect to \( x \) is given by:

• \( f'(x) = n \cdot x^{n-1} \)

Applying the power rule allows you to find derivatives quickly without using the limit definition every time. For the exercise's function \( f(x) = x^{-\frac{1}{2}} \), applying the power rule gives us directly:

• \( f'(x) = -\frac{1}{2} x^{-\frac{3}{2}} \)

The beauty of the power rule is in its simplicity and efficiency. Once you understand the rule, you can differentiate similar functions immediately. However, knowing how to derive it from the limit definition strengthens your grasp of calculus and prepares you for more complex problems.