Problem 61 - Calculus Volume 1

Question

Use the definition of a derivative to find \(f^{\prime}(x)\). $$ f(x)=\frac{9}{x} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{-9}{x^2} \).

Step 1: Understand the Definition of a Derivative

The derivative of a function \( f(x) \) at a point \( x \) is defined as \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \). This definition will be used to calculate the derivative of the given function \( f(x) = \frac{9}{x} \).

Step 2: Substitute the Function into the Derivative Formula

Substitute \( f(x) = \frac{9}{x} \) into the derivative formula. This gives:\[ f'(x) = \lim_{h \to 0} \frac{\frac{9}{x+h} - \frac{9}{x}}{h}. \]

Step 3: Simplify the Expression in the Numerator

Find a common denominator for \( \frac{9}{x+h} - \frac{9}{x} \). The common denominator is \( x(x+h) \), so:\[ \frac{9}{x+h} - \frac{9}{x} = \frac{9x - 9(x+h)}{x(x+h)} = \frac{-9h}{x(x+h)}. \]

Step 4: Substitute Back into the Derivative Formula

Substitute the simplified expression back into the derivative formula. This results in:\[ f'(x) = \lim_{h \to 0} \frac{-9h}{hx(x+h)}. \]

Step 5: Cancel \( h \) in the Numerator and Denominator

Cancel \( h \) in the numerator and the denominator:\[ f'(x) = \lim_{h \to 0} \frac{-9}{x(x+h)}. \]

Step 6: Evaluate the Limit as \( h \to 0 \)

Evaluate the limit by substituting \( h = 0 \):\[ f'(x) = \frac{-9}{x^2}. \] This is the derivative of the function using the definition of a derivative.

Key Concepts

Definition of a Derivative

The derivative of a function gives us the rate at which the function's value changes with respect to a change in its variable. In simpler terms, it tells us how steep the graph of a function is at any given point. The formal mathematical definition of a derivative is expressed as follows: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] In this equation, as the variable \( h \) approaches 0, the formula calculates the slope of the tangent line to the curve at the point \( x \). This concept is fundamental in calculus and is used to determine the behavior and trends of functions.

Limit Process

The limit process is crucial for understanding derivatives. It's essentially what allows us to pin down the exact slope of a function at a precise point. The limit process calculates what happens to a function as it approaches a particular value. When using the definition of a derivative, the limit \[ \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] ensures that you are zooming in on an infinitely small section of the graph to find the rate of change at point \( x \). This precise calculation depends on smoothly transitioning as \( h \) shrinks towards 0.

Rational Functions

Rational functions are functions that can be expressed as the ratio of two polynomials. A simple example is \( \frac{9}{x} \), where 9 is a constant polynomial and \( x \) is the denominator polynomial. These functions often include division by the variable, which can introduce vertical asymptotes. Rational functions have unique properties:

• They can be undefined where their denominator is zero.
• Their graphs may have horizontal and vertical asymptotes depending on the degrees of the polynomials involved.
• Finding derivatives of rational functions usually involves simplifying complex fractions by finding common denominators.

Understanding these characteristics is important when analyzing their derivatives.

Calculating Derivatives

Calculating derivatives involves using the definition of a derivative and often requires algebraic manipulation to simplify expressions. For the function \( f(x) = \frac{9}{x} \), its derivative came from carefully applying the limit process. Here's how:

• We started by substituting \( f(x+h) \) and \( f(x) \) into the definition of a derivative.
• Simplified the complex fraction to make the limit process easier by finding a common denominator.
• Reduced the fraction by canceling common terms, such as \( h \) in both the numerator and denominator.
• Evaluated the limit by letting \( h \to 0 \), resulting in a clean expression for the derivative: \( f'(x) = \frac{-9}{x^2} \).

This step-by-step reduction is essential in keeping calculus manageable and effective for finding precise rates of change.