Question

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

Short answer

Answer: The derivative of the function y = (x^2)/(1+x) with respect to x is y' = (2x + x^2)/((1+x)^2).

Step-by-step solution

Identify the functions u(x) and v(x)

In our case, \(y=\frac{x^2}{1+x}\). We can identify \(u(x) = x^2\) and \(v(x) = 1+x\).

Differentiate u(x) and v(x)

We need to differentiate \(u(x) = x^2\) and \(v(x) = 1+x\) with respect to \(x\). We get: $$u'(x) = 2x$$ $$v'(x) = 1$$

Apply the Quotient Rule

Now we can use the quotient rule to differentiate \(y = \frac{u(x)}{v(x)}\): $$y' = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$$ Substitute the expressions for \(u(x)\), \(v(x)\), \(u'(x)\), and \(v'(x)\): $$y' = \frac{(2x)(1+x) - (x^2)(1)}{(1+x)^2}$$

Simplify the expression

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

Key concepts

Quotient Rule

When dealing with functions that are expressed as one function divided by another, the quotient rule is a handy tool. It helps you differentiate these functions efficiently. The quotient rule states that if you have a function in the form \( y = \frac{u(x)}{v(x)} \), then the derivative \( y' \) can be found using the formula:

• \( y' = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \)

Here’s how you apply it:

• First, identify who is \( u(x) \) (the numerator) and who is \( v(x) \) (the denominator).
• Then, find the derivatives \( u'(x) \) and \( v'(x) \).
• Plug these values into the formula and simplify.

This technique is crucial because it allows us to calculate derivatives without having to multiply or divide large expressions directly.

Derivative

Derivatives are a foundational concept in calculus. They measure how a function changes as its input changes. In simpler terms, a derivative tells you the rate of change or the slope at any given point of a function. For polynomial functions, like \( x^2 \), finding a derivative is quite straightforward:

• Use the power rule which states that for \( x^n \), the derivative \( \frac{d}{dx}x^n = nx^{n-1} \).

In the exercise, we differentiated \( u(x) = x^2 \) to get \( u'(x) = 2x \), and \( v(x) = 1 + x \) to get \( v'(x) = 1 \). Differentiating each part correctly is key to applying the quotient rule successfully.
Derivatives help in understanding the behavior of functions, such as identifying maximum and minimum points, and predicting trends. They're widely used in fields ranging from physics to economics.

Simplifying Expressions

Once you have applied the quotient rule, you'll often end up with an expression that can be quite complex. Simplifying expressions makes them more manageable and easier to interpret. After applying the quotient rule in our exercise, we were left with:

• \( \frac{2x(1+x) - x^2}{(1+x)^2} \)

Let's break down the simplification:

• First, distribute the terms in the numerator: \( 2x + 2x^2 - x^2 \).
• Combine like terms: it simplifies to \( 2x + x^2 \).

The denominator \( (1+x)^2 \) remains as is. Simplifying the expression allows you to see the simplified relationship between variables, and it's crucial in reducing computational errors, especially in larger calculations.