Question

Differentiate the following functions: \(y=\frac{1}{1+x^{2}}\)

Short answer

Answer: The derivative of the function \(y=\frac{1}{1+x^2}\) with respect to x is \(y'=\frac{-2x}{(1+x^2)^2}\).

Step-by-step solution

Identify the numerator and denominator functions

In the given function, \(y=\frac{1}{1+x^{2}}\), we have: Numerator: \(u=1\) Denominator: \(v=1+x^2\)

Differentiate the numerator and denominator functions

Now, we differentiate \(u\) and \(v\) with respect to x: Derivative of the numerator: \(u'=\frac{d(u)}{d(x)}=\frac{d(1)}{d(x)}=0\) Derivative of the denominator: \(v'=\frac{d(v)}{d(x)}=\frac{d(1+x^2)}{d(x)}=0+2x=2x\)

Apply the quotient rule for differentiation

Using the quotient rule, we find the derivative of the function \(y\): \(y' = \frac{u'v-uv'}{v^2}\) Substitute the values of \(u\), \(v\), \(u'\), and \(v'\) into the formula: \(y' = \frac{(0)(1+x^2)-(1)(2x)}{(1+x^2)^2}\)

Simplify the expression

Now we simplify the expression for the derivative: \(y' = \frac{-2x}{(1+x^2)^2}\) So the derivative of the function \(y=\frac{1}{1+x^2}\) with respect to x is \(y'=\frac{-2x}{(1+x^2)^2}\).

Key concepts

Quotient Rule for Differentiation

Understanding the quotient rule for differentiation is essential when dealing with calculus problems involving rational functions, or functions that are expressed as a ratio of two polynomials.

The quotient rule is a method used to find the derivative of a function that is the quotient of two other functions. Given a function representable as \(y = \frac{u(x)}{v(x)}\) with \(u\) and \(v\) being differentiable functions of \(x\), the quotient rule states that the derivative \(y'\) of this function is given by:
\[y' = \frac{u'v - uv'}{v^2}\]
where \(u'\) is the derivative of \(u\) with respect to \(x\), and \(v'\) is the derivative of \(v\) with respect to \(x\). It's crucial to carefully apply each step, as skipping or miswriting one part can lead to an incorrect answer.

To put the quotient rule into practice, it's important to:

• Identify the top function (numerator) as \(u\) and the bottom function (denominator) as \(v\).
• Differentiate both \(u\) and \(v\) separately.
• Plug \(u\), \(v\), \(u'\), and \(v'\) into the quotient rule formula.
• Simplify the expression to find the final derivative.

This reliable step-by-step approach ensures a solid understanding and correct application of the quotient rule.

Calculus Problem Solving

Solving calculus problems effectively often requires a combination of analytical thinking and methodical application of mathematical rules. In the context of calculus problem solving, one generally goes through a series of steps to simplify the problem and arrive at the solution.

For instance, the process for differentiating a rational function could look like this:

• Identify the type of function and determine the rule that needs to be applied, such as the product rule, chain rule, or quotient rule of differentiation.
• Break down the function into simpler parts if necessary (e.g., separate the numerator and the denominator).
• Calculate the derivatives of individual components of the function.
• Apply the correct differentiation rule to combine these individual derivatives into the derivative of the overall function.
• Lastly, simplify the expression to the most simplified form for clarity and ease of use.

As you work through problems, remember to write each step clearly and revise your calculations to minimize errors. Keeping a systematic approach will help in understanding complex problems and breaking them down into manageable steps.

Derivative of a Function

The derivative of a function represents the rate at which the function's value changes with respect to a change in its input value, often thought of as the function's 'slope' or 'gradient'. In calculus, the concept of a derivative is fundamental and serves various applications in fields like physics, engineering, and economics.

To find the derivative of a function, one may apply different rules such as the power rule, product rule, chain rule, and quotient rule, among others. The choice of rule depends on the form of the function.

For example, to differentiate a simple power of \(x\), \(x^n\), we use the power rule: \( (x^n)' = nx^{n-1} \). However, more complex functions require a careful approach:

• First, analyze the function to identify its components and related rules for differentiation.
• Next, apply the appropriate rules to calculate the derivative.
• Finally, combine and simplify the results to obtain the final derivative.

Understanding how to differentiate various types of functions is a critical skill in calculus, opening the door to solving more advanced problems and building upon mathematical knowledge.