Problem 40 - Brief Applied Calculus

Question

Find \(f^{\prime}(x)\) by using the definition of the derivative. [Hint: See Example 4.] $$ \underline{\phantom{xxx}} f(x)=\frac{1}{x^{2}} $$

Step-by-Step Solution

Answer

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

Step 1: Recall the Definition of the Derivative

The definition of the derivative of a function \( f(x) \) at a point \( x \) is given by the limit: \[ f^{\prime}(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}. \] Here, \( h \) approaches zero, and we find the difference quotient of \( f(x) \).

Step 2: Substitute \( f(x) \) into the Definition

Substitute \( f(x) = \frac{1}{x^{2}} \) into the limit definition:\[ f(x+h) = \frac{1}{(x+h)^{2}}. \] Thus, the expression becomes:\[ f^{\prime}(x) = \lim_{{h \to 0}} \frac{\frac{1}{(x+h)^2} - \frac{1}{x^2}}{h}. \]

Step 3: Simplify the Difference Quotient

Simplify the expression:\[ \frac{1}{(x+h)^2} - \frac{1}{x^2} = \frac{x^2 - (x+h)^2}{(x+h)^2 x^2}. \] This simplifies further using the identity \((a^2 - b^2) = (a - b)(a + b)\):\[ = \frac{-(2xh + h^2)}{(x+h)^2 x^2}. \]

Step 4: Calculate the Limit

Insert the simplified difference into the limit:\[ f^{\prime}(x) = \lim_{{h \to 0}} \frac{-(2xh + h^2)}{h (x+h)^2 x^2}. \] Cancel \( h \) from the numerator and denominator:\[ = \lim_{{h \to 0}} \frac{-2x - h}{(x+h)^2 x^2}. \]As \( h \to 0 \), the expression simplifies to:\[ f^{\prime}(x) = \frac{-2x}{x^4} = \frac{-2}{x^3}. \]

Key Concepts

Limit Definition

Derivatives are a fundamental concept in calculus, representing the rate of change of a function. One standard method for finding a derivative at a point is using the limit definition. This approach is critical for understanding how derivatives truly capture the behavior of functions.

The limit definition of a derivative for a function \( f(x) \) at the point \( x \) is expressed as:

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

Here, \( h \) is a small change in \( x \), and the expression \( \frac{f(x+h) - f(x)}{h} \) is known as the difference quotient. As \( h \) approaches zero, we focus on how \( f(x) \) changes at that specific point. Calculating this limit helps us find the instantaneous rate of change or the slope of the tangent to the curve at \( x \). Understanding this definition lays the groundwork for unraveling the complexities of calculus effectively.

Difference Quotient

The difference quotient is pivotal for calculating derivatives using the limit approach. It gives us a way to express how the function value changes concerning a small change in the input variable.

The general form of the difference quotient is:

• \( \frac{f(x+h) - f(x)}{h} \)

When \( h \) is very small, this quotient approximates the slope of a line connecting two points on the function, allowing us to estimate changes. As \( h \) shrinks towards zero, the line becomes the tangent to the curve at \( x \), providing the exact slope at that point.

For the function \( f(x) = \frac{1}{x^2} \), substituting into the difference quotient gives us:

• \( \frac{\frac{1}{(x+h)^2} - \frac{1}{x^2}}{h} \)

This step is crucial for simplifying the expression before calculating the limit.

Rational Function Derivatives

Rational functions, those expressed as the quotient of two polynomials, have unique derivatives calculated using the same principles. The task involves simplifying expressions and applying the limit definition effectively, just as we did with the given function \( f(x) = \frac{1}{x^2} \).

To find the derivative of \( \frac{1}{x^2} \), we first simplify the difference quotient to make limit calculations manageable:

• \( \frac{\frac{1}{(x+h)^2} - \frac{1}{x^2}}{h} \) simplifies to \( \frac{-(2xh + h^2)}{h (x+h)^2 x^2} \)

By canceling \( h \) in the numerator and denominator, the expression further reduces, making it straightforward to compute the limit as \( h \to 0 \).

Through this process, we find that the derivative of a rational function involves polynomial simplifications and limit calculations. For \( f(x) = \frac{1}{x^2} \), the resulting derivative simplifies to \( f^{\prime}(x) = \frac{-2}{x^3} \). Understanding these steps is fundamental for mastering derivative calculations for any rational function.