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Chapter 3: Problem 66

Use any method to evaluate the derivative of the following functions. $$f(x)=\frac{4-x^{2}}{x-2}$$

Short Answer

Expert verified
Question: Find the derivative of the given function $$f(x) = \frac{4-x^2}{x-2}$$. Answer: The derivative of the given function is $$f'(x) = \frac{-x^2+4x-4}{(x-2)^2}$$.

Step by step solution

01

Identify the Basic Functions

In our case, the function $$f(x) = \frac{4-x^2}{x-2}$$ can be broken down into two basic functions: $$g(x) = 4-x^2$$ $$k(x) = x-2$$
02

Differentiate each Basic Function

We next need to find the derivatives of both $$g(x)$$ and $$k(x)$$. The derivative of $$g(x)$$ is: $$g'(x) = \frac{d}{dx}(4-x^2) = -2x$$ The derivative of $$k(x)$$ is: $$k'(x) = \frac{d}{dx}(x-2) = 1$$
03

Apply the Quotient Rule

We now apply the quotient rule. Using the formula $$h'(x) = \frac{g'(x)k(x) - g(x)k'(x)}{(k(x))^2},$$ we plug in the values of $$g(x), k(x), g'(x)$$, and $$k'(x)$$ to obtain: $$f'(x) = \frac{(-2x)(x-2) - (4-x^2)(1)}{((x-2))^2}$$
04

Simplify the Result

Lastly, we simplify the expression as much as possible: $$f'(x) = \frac{-2x^2+4x - 4 +x^2}{(x-2)^2}$$ Combine the similar terms: $$f'(x) = \frac{-x^2+4x-4}{(x-2)^2}$$ Thus, the derivative of the given function is: $$f'(x) = \frac{-x^2+4x-4}{(x-2)^2}$$

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Key Concepts

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

Understanding the Quotient Rule
The quotient rule is a handy tool in calculus, specifically used for finding the derivative of a division of two functions. Suppose you have a function \( f(x) = \frac{g(x)}{k(x)} \). The quotient rule allows you to differentiate \( f(x) \) by utilizing the derivatives of the numerator \( g(x) \) and the denominator \( k(x) \). To do this, you would apply the formula:
  • \( f'(x) = \frac{g'(x)k(x) - g(x)k'(x)}{(k(x))^2} \)
This formula might look complex, but breaking it down makes it easier to apply. It essentially states that:
  • First, multiply the derivative of the numerator \( g'(x) \) by the denominator \( k(x) \).
  • Secondly, subtract the product of the numerator \( g(x) \) and the derivative of the denominator \( k'(x) \) from the first step.
  • Finally, divide the result by the square of the denominator \( (k(x))^2 \).
The quotient rule is essential for functions that are fractions, enabling us to assess rates of change in real-world situations like speed or density.
Exploring Differentiation
Differentiation is a fundamental concept in calculus that deals with finding the rate at which one quantity changes about another. When we differentiate a function, we essentially find its slope at any given point. Consider a function \( y = g(x) \); its derivative, denoted as \( g'(x) \), represents the rate of change of \( y \) with respect to \( x \).
Learning the basic differentiation rules is crucial:
  • The power rule states that the derivative of \( x^n \) is \( nx^{n-1} \).
  • The constant rule makes it clear that the derivative of a constant is zero.
  • The sum rule allows you to add the derivatives of functions that are added together.
In our exercise, differentiating \( g(x) = 4-x^2 \) using the power rule, we get \( g'(x) = -2x \). Similarly, differentiating \( k(x) = x-2 \) yields \( k'(x) = 1 \). Practicing these rules enhances our ability to solve calculus problems seamlessly.
The Art of Simplifying Expressions
Simplifying expressions is a necessary skill in calculus to make derivatives easier to interpret and solve. After applying rules like the quotient rule, expressions can become quite bulky, so we must reduce them to the simplest form possible.
The simplification process involves:
  • Combining like terms, which means summing terms that have the same variable raised to the same power. For instance, \(-2x^2 + x^2\) can be simplified to \(-x^2\).
  • Cancelling common factors, which can sometimes occur in both the numerator and denominator.
  • Reorganizing terms to ensure that the expression is neat and concise.
In our exercise, after differentiating, the expression \( f'(x) = \frac{-2x^2+4x - 4 +x^2}{(x-2)^2} \) is simplified by combining like terms to \( \frac{-x^2+4x-4}{(x-2)^2} \). This simplification not only makes the solution easier to understand but also helps in verifying that our differentiation steps were carried out correctly and efficiently.

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Most popular questions from this chapter

Let \(C(x)\) represent the cost of producing \(x\) items and \(p(x)\) be the sale price per item if \(x\) items are sold. The profit \(P(x)\) of selling x items is \(P(x)=x p(x)-C(x)\) (revenue minus costs). The average profit per item when \(x\) items are sold is \(P(x) / x\) and the marginal profit is dP/dx. The marginal profit approximates the profit obtained by selling one more item given that \(x\) items have already been sold. Consider the following cost functions \(C\) and price functions \(p\). a. Find the profit function \(P\). b. Find the average profit function and marginal profit function. c. Find the average profit and marginal profit if \(x=a\) units are sold. d. Interpret the meaning of the values obtained in part \((c)\). $$C(x)=-0.02 x^{2}+50 x+100, p(x)=100, a=500$$

Suppose you own a fuel-efficient hybrid automobile with a monitor on the dashboard that displays the mileage and gas consumption. The number of miles you can drive with \(g\) gallons of gas remaining in the tank on a particular stretch of highway is given by \(m(g)=50 g-25.8 g^{2}+12.5 g^{3}-1.6 g^{4},\) for \(0 \leq g \leq 4\). a. Graph and interpret the mileage function. b. Graph and interpret the gas mileage \(m(g) / \mathrm{g}\). c. Graph and interpret \(d m / d g\).

Use the properties of logarithms to simplify the following functions before computing \(f^{\prime}(x)\). $$f(x)=\ln \sqrt{10 x}$$,

General logarithmic and exponential derivatives Compute the following derivatives. Use logarithmic differentiation where appropriate. $$\frac{d}{d x}\left(1+x^{2}\right)^{\sin x}$$

A 500-liter (L) tank is filled with pure water. At time \(t=0,\) a salt solution begins flowing into the tank at a rate of \(5 \mathrm{L} / \mathrm{min} .\) At the same time, the (fully mixed) solution flows out of the tank at a rate of \(5.5 \mathrm{L} / \mathrm{min}\). The mass of salt in grams in the tank at any time \(t \geq 0\) is given by $$M(t)=250(1000-t)\left(1-10^{-30}(1000-t)^{10}\right)$$ and the volume of solution in the tank (in liters) is given by \(V(t)=500-0.5 t\). a. Graph the mass function and verify that \(M(0)=0\). b. Graph the volume function and verify that the tank is empty when \(t=1000\) min. c. The concentration of the salt solution in the tank (in \(\mathrm{g} / \mathrm{L}\) ) is given by \(C(t)=M(t) / V(t) .\) Graph the concentration function and comment on its properties. Specifically, what are \(C(0)\) and \(\lim _{\theta \rightarrow 000^{-}} C(t) ?\) \(t \rightarrow 1\) d. Find the rate of change of the mass \(M^{\prime}(t),\) for \(0 \leq t \leq 1000\). e. Find the rate of change of the concentration \(C^{\prime}(t),\) for \(0 \leq t \leq 1000\). f. For what times is the concentration of the solution increasing? Decreasing?
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