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Chapter 2: Problem 7

Use the Quotient Rule to differentiate the function. \(f(x)=\frac{x}{x^{2}+1}\)

Short Answer

Expert verified
The derivative of the function \(f(x)=\frac{x}{x^{2}+1}\) using the Quotient Rule is \(f'(x) = \frac{1-x^{2}}{(x^{2}+1)^{2}}\).

Step by step solution

01

Identify the functions

First, identify the functions in the quotient. Here, the numerator function \(u(x)\) is \(x\) and the denominator function \(v(x)\) is \(x^{2} + 1\).
02

Differentiate the functions

Take the derivative of \(u(x)\) and \(v(x)\). The derivative of \(u = x\) is \(u' = 1\) and derivative of \(v = x^{2} + 1\) is \(v' = 2x\).
03

Apply the Quotient Rule

Apply the Quotient Rule formula which is \(\frac{v \cdot u' - u \cdot v'}{{v}^{2}}\). Substituting the known values, get \(\frac{(x^{2}+1) \cdot 1 - x \cdot 2x}{(x^{2}+1)^{2}}\). This simplifies to \(\frac{-x^{2}+1}{(x^{2}+1)^{2}}\)
04

Simplify the result

The function can be simplified as \(\frac{1-x^{2}}{(x^{2}+1)^{2}}\) which is the derivative of the function.

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Key Concepts

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

Calculus
Calculus is a branch of mathematics that deals with the study of change and motion. Through its two major subdivisions, differential calculus and integral calculus, it enables us to understand and describe the dynamic world.

Differential calculus is concerned with the concept of a derivative, a measure of how a function changes as its input changes. Essentially, it's about slopes of curves and rates of change. Integral calculus, on the other hand, focuses on the accumulation of quantities, such as areas under curves and the total change from one point to another.

Using calculus, we can solve problems that involve complex shapes, chaotic motion, and changing forces. These issues arise throughout scientific and engineering domains, making calculus an essential tool for a variety of fields.
Derivative of a Function
The derivative of a function at a point is the rate at which the function's value changes with respect to a change in its input value. Simply put, it's how fast the function is moving at that point. When represented graphically, the derivative corresponds to the slope of the tangent line at a point on a curve.

In practical terms, if you have a function representing the position of a car over time, the derivative will tell you the car鈥檚 velocity 鈥� how fast the car is going at any given moment. To find the derivative, we use various rules and formulas that differ depending on the form of the function being differentiated. The derivative of a function is denoted by 'f'(x) (read as

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Most popular questions from this chapter

Prove (Theorem 2.3) that \(\frac{d}{d x}\left[x^{n}\right]=n x^{n-1}\) for the case in which \(n\) is a rational number. (Hint: Write \(y=x^{p / q}\) in the form \(y^{q}=x^{p}\) and differentiate implicitly. Assume that \(p\) and \(q\) are integers, where \(q>0 .\) )

The radius \(r\) of a circle is increasing at a rate of 3 centimeters per minute. Find the rates of change of the area when (a) \(r=6\) centimeters and (b) \(r=24\) centimeters.

A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep.

The formula for the volume of a cone is \(V=\frac{1}{3} \pi r^{2} h .\) Find the rate of change of the volume if \(d r / d t\) is 2 inches per minute and \(h=3 r\) when (a) \(r=6\) inches and (b) \(r=24\) inches.

Find \(d y / d x\) implicitly and find the largest interval of the form \(-a
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