Question

Use \(f^{\prime}(x)=\lim _{h \rightarrow 0}[f(x+h)-f(x)] / h\) to find the derivative at \(x\). $$ H(x)=\frac{3}{\sqrt{x-2}} $$

Short answer

The derivative is \(H'(x) = \frac{-3}{(x-2)^{3/2}}\).

Step-by-step solution

Define the Difference Quotient

To find the derivative of the function \( H(x) = \frac{3}{\sqrt{x-2}} \), we need to begin by applying the definition of the derivative. The difference quotient for the function is \( \frac{H(x+h) - H(x)}{h} \).

Substitute the Function into the Difference Quotient

Substitute \( H(x) = \frac{3}{\sqrt{x-2}} \) and \( H(x+h) = \frac{3}{\sqrt{x+h-2}} \) into the difference quotient. The expression becomes: \[ \frac{\frac{3}{\sqrt{x+h-2}} - \frac{3}{\sqrt{x-2}}}{h} \]

Combine the Fractions

To combine the fractions in the numerator, find a common denominator, which is \(\sqrt{x+h-2} \cdot \sqrt{x-2}\). The numerator becomes:\[ \frac{3\sqrt{x-2} - 3\sqrt{x+h-2}}{\sqrt{x+h-2} \cdot \sqrt{x-2}} \]

Simplify the Expression

Now, the expression for the derivative is: \[ \frac{3\sqrt{x-2} - 3\sqrt{x+h-2}}{h \cdot \sqrt{x+h-2} \cdot \sqrt{x-2}} \]Factor out a 3 and multiply numerator and denominator by the conjugate:Multiply by \(\sqrt{x-2} + \sqrt{x+h-2}\) over itself to eliminate the square root.

Rationalize the Numerator

After multiplying by the conjugate, the numerator becomes:\[ 3((x-2) - (x+h-2)) = 3(-h)\]The simplified expression is:\[ \frac{-3h}{h(\sqrt{x-2} \cdot \sqrt{x+h-2} (\sqrt{x-2} + \sqrt{x+h-2}))} \]

Cancel and Simplify

Cancel the \(h\) from the numerator and denominator:\[ \frac{-3}{\sqrt{x-2} \cdot \sqrt{x+h-2} (\sqrt{x-2} + \sqrt{x+h-2})} \]

Evaluate the Limit

Take the limit as \(h \to 0\):\[ \lim _{h \rightarrow 0} \frac{-3}{\sqrt{x-2} \cdot \sqrt{x+h-2} (\sqrt{x-2} + \sqrt{x+h-2})} \]As \(h\) approaches zero, \(\sqrt{x+h-2}\) approaches \(\sqrt{x-2}\). The expression simplifies to:\[ \frac{-3}{(x-2)^{3/2}} \]

Conclusion: Write the Derivative

Thus, the derivative of the function \( H(x) = \frac{3}{\sqrt{x-2}} \) is given by:\(H'(x) = \frac{-3}{(x-2)^{3/2}}\)

Key concepts

Difference Quotient

The difference quotient is a foundational concept in calculus used to define the derivative. It's expressed as \( \frac{f(x+h) - f(x)}{h} \), which represents the average rate of change of the function \( f \) over an interval \( h \). This formula evaluates the slope of the secant line between two points on the graph of \( f \). By calculating this quotient, we can explore how the function changes as we move through different values of \( x \) with an increment of \( h \).
In our exercise, to find the derivative of \( H(x) = \frac{3}{\sqrt{x-2}} \), the function is plugged into the difference quotient: \( \frac{\frac{3}{\sqrt{x+h-2}} - \frac{3}{\sqrt{x-2}}}{h} \). With this setup, we prepare to explore the limit process that will help identify the derivative at a particular point.

Limit Process in Calculus

The limit process is crucial in transitioning from average to instantaneous rates of change. In calculus, taking the limit as \( h \to 0 \) of the difference quotient allows us to find the exact slope at a single point, which is the derivative.
After setting up the difference quotient for the function \( H(x) \), we aim to find \( \lim_{h \to 0} \frac{H(x+h) - H(x)}{h} \). This transformation presents challenges, such as handling indeterminate forms like \( \frac{0}{0} \). We must manipulate the expression so that the limit provides a precise value. As \( h \) approaches zero, the expression often simplifies, enabling us to calculate the derivative.

Rationalizing Denominators

Rationalizing denominators involves simplifying expressions to eliminate the square roots in denominators. This is particularly important when working with complex fractions in the derivative calculation.
In our exercise, to rationalize the complex expression \( \frac{3\sqrt{x-2} - 3\sqrt{x+h-2}}{h \cdot \sqrt{x+h-2} \cdot \sqrt{x-2}} \), we multiply both numerator and denominator by the conjugate \( \sqrt{x-2} + \sqrt{x+h-2} \). This step simplifies square root expressions and facilitates further simplifications, such as canceling out common terms. It also paves the way to evaluate the limit efficiently, leading to the final derivative.

Finding Derivatives Step by Step

Finding derivatives involves several strategic steps to ensure accuracy. It starts with setting up the difference quotient and follows with careful application of the limit process, while sometimes requiring algebraic manipulation like rationalizing denominators.
In our given solution, the steps began with identifying the difference quotient for \( H(x) = \frac{3}{\sqrt{x-2}} \). After substituting the function, algebraic techniques helped simplify the expression, particularly through rationalization, ultimately allowing the calculation of the limit. This step-by-step process concluded by effectively deriving \( H'(x) \) as \( \frac{-3}{(x-2)^{3/2}} \), showcasing both the importance of understanding each stage and the interconnectedness of these mathematical concepts.