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Chapter 3: Problem 25

Use the formula $$f^{\prime}(x)=\lim _{z \rightarrow x} \frac{f(z)-f(x)}{z-x}$$ to find the derivative of the functions $$g(x)=\frac{x}{x-1}$$

Short Answer

Expert verified
The derivative of \( g(x) = \frac{x}{x-1} \) is \( g^{\prime}(x) = \frac{-1}{(x-1)^2} \).

Step by step solution

01

Identify the Function and Apply the Formula

We are given the function \( g(x) = \frac{x}{x-1} \) and need to find its derivative using the definition of the derivative. According to the formula: \[ f^{\prime}(x)=\lim _{z \rightarrow x} \frac{f(z)-f(x)}{z-x} \] we substitute \( f(x) = g(x) = \frac{x}{x-1} \) into this equation.
02

Substitute and Simplify

First, calculate \( g(z) \) as \( \frac{z}{z-1} \). Then set up the equation: \[ g^{\prime}(x) = \lim _{z \rightarrow x} \frac{\frac{z}{z-1} - \frac{x}{x-1}}{z-x} \]Simplify the expression: find a common denominator for the fractions in the numerator, which gives:\[ \frac{z(x-1) - x(z-1)}{(z-1)(x-1)} \]
03

Further Simplification

Distribute and simplify the numerators:\[ z(x-1) = zx - z \]\[ x(z-1) = xz - x \]Plug these back into the numerator:\[ \frac{zx - z - xz + x}{(z-1)(x-1)} = \frac{x - z}{(z-1)(x-1)} \]
04

Cancel and Solve the Limit

Notice in the expression:\[ \lim _{z \rightarrow x} \frac{x - z}{(z-1)(x-1)} \]That \( x - z \) can initially be written as \(-(z-x)\). Hence, we rewrite the fraction:\[ \frac{-(z-x)}{(z-1)(x-1)} \]Cancel \( z-x \) from the numerator and denominator, which leaves:\[ \lim _{z \rightarrow x} \frac{-1}{(z-1)(x-1)} \]Evaluate the limit as \( z \to x \):\[ \frac{-1}{(x-1)^2} \]
05

State the Derivative

Finally, the derivative of \( g(x) = \frac{x}{x-1} \) is \[ g^{\prime}(x) = \frac{-1}{(x-1)^2} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limit
In calculus, a limit is a fundamental concept that describes the behavior of a function as the input approaches a particular point. When finding the derivative, limits help us understand how a function is changing at an extremely small scale. More specifically, we examine how the function behaves as it gets infinitely close to a given point. This process is formalized in the derivative formula:\[ f'(x) = \lim_{z \rightarrow x} \frac{f(z) - f(x)}{z - x} \]In this expression, as \( z \) approaches \( x \), the difference between \( f(z) \) and \( f(x) \) helps determine the slope of the tangent line at that point. By evaluating this limit, we can precisely calculate the rate of change or the slope of the function at any given point along its curve. It's essential to correctly simplify and evaluate limits to find accurate derivatives when dealing with more intricate functions.
Difference Quotient
The difference quotient is a method used to approximate the derivative of a function. It measures the average rate of change of the function over a small interval and helps in understanding the concept of the derivative. The difference quotient is given by the formula:\[ \frac{f(z) - f(x)}{z - x} \]This expression calculates the slope of the secant line through two points on the graph of a function. As \( z \) approaches \( x \), the secant line becomes the tangent line, providing us with the instantaneous rate of change or the derivative. Solving the difference quotient involves the following steps:
  • Substitute the function into the formula.
  • Simplify the resulting expression, often by finding a common denominator or combining similar terms.
  • Find the limit, which can sometimes involve canceling factors from the numerator and denominator.
Mathematical simplification is often necessary to make the evaluation of limits possible, especially when dealing with more complex functions like rational functions.
Rational Function
A rational function is a type of function that can be represented as the ratio of two polynomials. These functions take the form:\[ g(x) = \frac{p(x)}{q(x)} \]where \( p(x) \) and \( q(x) \) are polynomials. Rational functions can have points where they are not defined, often termed as discontinuities or holes due to division by zero. In this context, specific care must be taken to properly handle these points, especially when finding derivatives.To find the derivative of a rational function using limits, as we did with the function \( g(x) = \frac{x}{x-1} \), the process involves setting up the difference quotient and simplifying it. Key steps include finding common denominators, simplifying expressions, and canceling terms that contribute to discontinuities.It's important to note that when taking the derivative of rational functions, certain features like horizontal and vertical asymptotes play a critical role in understanding its behavior. Successful evaluation of the derivative requires careful handling of these traits together with limit processes.

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Most popular questions from this chapter

Use a CAS to perform the following steps for the functions \begin{equation} \begin{array}{l}{\text { a. Plot } y=f(x) \text { to see that function's global behavior. }} \\ {\text { b. Define the difference quotient } q \text { at a general point } x, \text { with }} \\ {\text { general step size } h .} \\\ {\text { c. Take the limit as } h \rightarrow 0 . \text { What formula does this give? }} \\ {\text { d. Substitute the value } x=x_{0} \text { and plot the function } y=f(x)} \\ \quad {\text { together with its tangent line at that point. }} \\ {\text { e. Substitute various values for } x \text { larger and smaller than } x_{0} \text { into }} \\ \quad {\text { the formula obtained in part (c). Do the numbers make sense }} \\ \quad {\text { with your picture? }} \\ {\text { f. Graph the formula obtained in part (c). What does it mean }} \\ \quad {\text { when its values are negative? Zero? Positive? Does this make }} \\ \quad {\text { sense with your plot from part (a)? Give reasons for your }} \\ \quad {\text { answer. }}\end{array} \end{equation} $$f(x)=\frac{x-1}{3 x^{2}+1}, \quad x_{0}=-1$$

a. Find the tangent to the curve \(y=2 \tan (\pi x / 4)\) at \(x=1\) b. Slopes on a tangent curve What is the smallest value the slope of the curve can ever have on the interval \(\quad-2

If \(x^{2} y^{3}=4 / 27\) and \(d y / d t=1 / 2,\) then what is \(d x / d t\) when \(x=2 ?\)

A building's shadow On a morning of a day when the sun will pass directly overhead, the shadow of an 80-ft building on level ground is 60 \(\mathrm{ft}\) long. At the moment in question, the angle \(\theta\) the sun makes with the ground is increasing at the rate of \(0.27^{\circ} / \mathrm{min}\) . At what rate is the shadow decreasing? (Remember to use radians. Express your answer in inches per minute, to the nearest tenth.)

Clock's moving hands At what rate is the angle between a clock's minute and hour hands changing at 4 o'clock in the afternoon?
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