Problem 63 - Calculus Volume 1

Question

Use the definition of a derivative to find \(f^{\prime}(x)\). $$ f(x)=\frac{1}{\sqrt{x}} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Definition of the Derivative

Using the limit definition, the derivative of a function \(f(x)\) is given by \(f^{\prime}(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}\). We will apply this to \(f(x)=\frac{1}{\sqrt{x}}\).

Step 2: Substitute into the Derivative Definition

First, substitute \(f(x+h)\) and \(f(x)\) into the definition. This gives us: \(f(x+h) = \frac{1}{\sqrt{x+h}}\) and \(f(x) = \frac{1}{\sqrt{x}}\).

Step 3: Write the Expression for the Difference Quotient

Substitute the expressions of \(f(x+h)\) and \(f(x)\) into the difference quotient: \(\frac{f(x+h) - f(x)}{h} = \frac{\frac{1}{\sqrt{x+h}} - \frac{1}{\sqrt{x}}}{h}\).

Step 4: Simplify the Difference Quotient

To simplify, we need to combine the fractions in the numerator. Use a common denominator, \(\sqrt{x}\cdot\sqrt{x+h}\): \[\frac{1}{\sqrt{x+h}} - \frac{1}{\sqrt{x}} = \frac{\sqrt{x} - \sqrt{x+h}}{\sqrt{x}\sqrt{x+h}}\].

Step 5: Multiply and Simplify Further

Substitute back into the difference quotient:\[\frac{\sqrt{x} - \sqrt{x+h}}{h\sqrt{x}\sqrt{x+h}}\]. Multiply by the conjugate: \(\frac{\sqrt{x} - \sqrt{x+h}}{h} \cdot \frac{\sqrt{x} + \sqrt{x+h}}{\sqrt{x} + \sqrt{x+h}}\).

Step 6: Calculate the Limit as h Approaches 0

This multiplication results in:\[ \frac{(\sqrt{x})^2 - (\sqrt{x+h})^2}{h(\sqrt{x}\sqrt{x+h})(\sqrt{x} + \sqrt{x+h})} \]which becomes:\[ \frac{x - (x+h)}{h(\sqrt{x}\sqrt{x+h})(\sqrt{x} + \sqrt{x+h})} \]Simplify this to:\[ \frac{-h}{h(\sqrt{x}\sqrt{x+h})(\sqrt{x} + \sqrt{x+h})} \].

Step 7: Tend h Towards Zero and Solve

Cancel \(h\) in the expression:\[ \frac{-1}{\sqrt{x}\sqrt{x+h}(\sqrt{x} + \sqrt{x+h})} \]As \(h\) approaches 0, the expression becomes:\[ \frac{-1}{x\cdot 2\sqrt{x}} = \frac{-1}{2x^{3/2}}\].

Step 8: Format the Final Derivative

We conclude that the derivative of \(f(x)\) is given by:\[ f^{\prime}(x) = \frac{-1}{2x^{3/2}} \].

Key Concepts

Limit Definition

The limit definition of a derivative is a fundamental concept in calculus. This definition is represented by the equation:

• \(f^{\prime}(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}\)

This expression is used to find the slope of the tangent line to the function at any point \(x\).

By applying this definition, we can determine how a function is changing at a small interval around \(x\). The basic idea is to take the difference of the function values at two points, divided by the distance between these points (\(h\)), and then evaluate as \(h\) approaches zero.For the function \(f(x)=\frac{1}{\sqrt{x}}\), using the limit definition allows us to derive its derivative—which tells us the rate at which the function's output is changing as \(x\) changes.

Difference Quotient

The difference quotient is a key concept in the limit definition of the derivative. It is given by the formula:

• \(\frac{f(x+h) - f(x)}{h}\)

This expression essentially calculates the average rate of change of the function over the interval from \(x\) to \(x+h\).

In our example, substituting \(f(x+h) = \frac{1}{\sqrt{x+h}}\) and \(f(x) = \frac{1}{\sqrt{x}}\) into the difference quotient gives us:

• \(\frac{\frac{1}{\sqrt{x+h}} - \frac{1}{\sqrt{x}}}{h}\)

This expression allows us to analyze the behavior of \(f(x)\) over small variations \(h\), ultimately helping us find the instantaneous rate of change by letting \(h\) approach zero.

Simplification Steps

Simplifying the difference quotient is an important step in finding the derivative. It often involves algebraic manipulation to make it easier to evaluate the limit.

Combining Fractions

Combining fractions, such as \(\frac{1}{\sqrt{x+h}}\) and \(\frac{1}{\sqrt{x}}\), requires finding a common denominator. In this case, the common denominator is \(\sqrt{x}\sqrt{x+h}\), leading to:

• \(\frac{\sqrt{x} - \sqrt{x+h}}{\sqrt{x}\sqrt{x+h}}\)

This makes the numerator simpler to manage.

Reduction

After substituting back, the entire expression becomes:

• \(\frac{\sqrt{x} - \sqrt{x+h}}{h\sqrt{x}\sqrt{x+h}}\)

These simplification steps are crucial before applying any further methods like the conjugate method to solve the problem fully.

Conjugate Method

The conjugate method is a valuable technique to simplify expressions involving square roots. It involves multiplying the expression by a form of 1—in this case, the conjugate of the numerator.

Using the Conjugate

For the expression \(\sqrt{x} - \sqrt{x+h}\), the conjugate is \(\sqrt{x} + \sqrt{x+h}\). Multiplying by the conjugate helps eliminate the square roots in the numerator, which is often necessary for evaluating limits.

• \(\frac{\sqrt{x} - \sqrt{x+h}}{h} \cdot \frac{\sqrt{x} + \sqrt{x+h}}{\sqrt{x} + \sqrt{x+h}}\)

This multiplication simplifies to:

• \(\frac{(\sqrt{x})^2 - (\sqrt{x+h})^2}{h(\sqrt{x}\sqrt{x+h})(\sqrt{x} + \sqrt{x+h})}\)

Resulting Simplification

Upon simplification, this step helps us more easily cancel out \(h\) from the numerator and denominator, allowing for the straightforward evaluation of the limit as \(h\) approaches zero.

• Ultimately, this results in a clean expression for the derivative: \(\frac{-1}{2x^{3/2}}\).

The conjugate method is not only a powerful tool for derivatives involving square roots but also a strategy used in rationalizing many algebraic expressions.