Question

Differentiate the following funetions: \(u=\frac{x}{\sqrt{1+x+x^{2}}}\)

Short answer

Answer: The derivative of the function is \(u'(x) = \frac{2\sqrt{1+x+x^2} - x(1+2x)}{2\sqrt{1+x+x^2}(1+x+x^2)}\).

Step-by-step solution

Find the derivatives of f(x) and g(x)

To differentiate \(f(x) = x\), we have: \(f'(x) = 1\) To differentiate \(g(x) = \sqrt{1+x+x^2}\), we see that \(g(x) = (1+x+x^2)^{\frac{1}{2}}\). So, \(g'(x) = \frac{1}{2}(1+x+x^2)^{-\frac{1}{2}}\frac{d}{dx}(1+x+x^2)\) The derivative of the inside function (1+x+x^2) is: \(\frac{d}{dx}(1+x+x^2) = 0+1+2x\) Putting this back into the expression for \(g'(x)\), we get: \(g'(x) = \frac{1}{2}(1+x+x^2)^{-\frac{1}{2}}(1+2x)\)

Apply the quotient rule

Now, we can apply the quotient rule to the function \(u(x) = \frac{x}{\sqrt{1+x+x^2}}\). Using our results from Step 1: \(u'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}\) Substitute \(f'(x)\), \(g(x)\), \(f(x)\), and \(g'(x)\): \(u'(x) = \frac{1 \cdot \sqrt{1+x+x^2} - x \cdot \frac{1}{2}(1+x+x^2)^{-\frac{1}{2}}(1+2x)}{(1+x+x^2)}\)

Simplify the result

To simplify the derivative, notice that the first term of the numerator has \(\sqrt{1+x+x^2}\), and we can rewrite the second term as \(\frac{-x(1+2x)}{2\sqrt{1+x+x^2}}\). So we have: \(u'(x) = \frac{\sqrt{1+x+x^2} - \frac{x(1+2x)}{2\sqrt{1+x+x^2}}}{(1+x+x^2)}\) Now, combine the terms in the numerator by finding a common denominator and combine fractions: \(u'(x) = \frac{2\sqrt{1+x+x^2} - x(1+2x)}{2\sqrt{1+x+x^2}(1+x+x^2)}\) So, the derivative of the given function is: \(u'(x) = \frac{2\sqrt{1+x+x^2} - x(1+2x)}{2\sqrt{1+x+x^2}(1+x+x^2)}\)

Key concepts

Quotient Rule

The Quotient Rule is a handy tool in calculus for differentiating functions that are divided by each other. When you have a function of the form \(u(x) = \frac{f(x)}{g(x)}\), the derivative can be found using the formula:

• \(u'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}\)

To apply the Quotient Rule, it's crucial to first differentiate both the numerator \(f(x)\) and the denominator \(g(x)\) separately. Then, substitute these derivatives back into the Quotient Rule formula. Be careful with the order of operations—subtract \(f(x)g'(x)\) from \(f'(x)g(x)\) to get the proper result. The denominator is squared to ensure the result always remains defined as long as \(g(x) eq 0\).
Practicing this process will help in handling more complicated functions, like the one given by \(u(x) = \frac{x}{\sqrt{1+x+x^2}}\), where precise substitution and simplification of terms are needed.

Chain Rule

The Chain Rule is essential when dealing with composite functions, where one function is nested inside another. If you have a function \(y = g(f(x))\), the Chain Rule lets you differentiate it by taking the derivative of the outer function \(g\) with respect to the inner function \(f\), and then multiplying it by the derivative of the inner function \(f(x)\). It's represented as:

• \(\frac{dy}{dx} = g'(f(x)) \cdot f'(x)\)

In the provided exercise, the Chain Rule helps differentiate \(g(x) = \sqrt{1+x+x^2}\) by treating it as \((1+x+x^2)^{\frac{1}{2}}\). To find its derivative \(g'(x)\), we start by applying the power rule to the outer function and then the derivative of the inside expression \(1+x+x^2\) using basic rules of differentiation.
Ensuring mastery of the Chain Rule is critical for effectively tackling problems where functions are compounded or not expressed in standard form.

Differentiation Steps

Differentiation is the process of finding the derivative of a function, which represents the rate of change. There are several steps and rules we can follow to simplify our work, especially for complex functions. To differentiate a function like \(u(x) = \frac{x}{\sqrt{1+x+x^2}}\), start by breaking down the problem:

• Differentiate the individual components. Use the basic rules to find \(f'(x)\) and \(g'(x)\).
• Apply specialized tools like the Quotient Rule and Chain Rule as needed for the given function structure. Each rule has its formula and method to transform terms properly.
• Simplify the resulting expression by factoring, combining like terms, and reducing fractions to make the final derivative as clean as possible.

Using these steps in order, and understanding why you do each one, will turn differentiation from a daunting task into a methodical process. Always being careful with algebra and function manipulation ensures you're making the correct moves at each step.