Question

Find the derivative of the algebraic function. $$ f(x)=\frac{x^{3}+3 x+2}{x^{2}+1} $$

Short answer

Thus, the derivative of the given function \(f(x)\) is \(f'(x) = \frac{2x^3 + 6x^2 + 6x - 2x^4 - 6x^3}{(x^2 + 1)^2}\).

Step-by-step solution

Identify u and v

Given that \( f(x) = \frac{x^{3}+3x+2}{x^{2}+1} \), let's assign \(u\) as \(x^3 + 3x + 2\) and \(v\) as \(x^2 + 1\).

Derive u and v

Apply the power rule to find the derivative of \(u\) and \(v\). Hence, \(u' = 3x^2 + 3\) and \(v' = 2x\).

Apply the quotient rule

Now, substitute \(u\), \(v\), \(u'\), and \(v'\) into the quotient rule equation: \((u/v)' = (v \cdot u' - u \cdot v') / v^2\). After substitution, we get \((x^{3}+3x+2)' = ((x^{2} + 1) \cdot (3x^2 + 3) - (x^3 + 3x + 2) \cdot 2x) / (x^{2} + 1)^2 \).

Simplify the equation

Simplify the above equation to get the final answer. After simplification, you should get: \(f'(x) = \frac{2x^3 + 6x^2 + 6x - 2x^4 - 6x^3}{(x^2 + 1)^2}\).

Key concepts

Quotient Rule

When it comes to finding the derivative of a function that is the ratio of two differentiable functions, the quotient rule is essential. Imagine the situation where you're sharing a pizza (the numerator) and are trying to determine how changing the number of friends (the denominator) affects everyone's share. That's where the quotient rule comes into play. In mathematical terms, for two functions, we express them as \( u(x) \) and \( v(x) \), where \( v(x) eq 0 \).

The quotient rule is: \[ (\frac{u}{v})' = \frac{vu' - uv'}{v^2} \]. Here, \( u' \) and \( v' \) are the derivatives of \( u(x) \) and \( v(x) \) respectively. This formula is crucial when dealing with complex functions like the one presented in our exercise: \( f(x)=\frac{x^{3}+3 x+2}{x^{2}+1} \). By identifying and differentiating the numerator (\( u \)) and the denominator (\( v \)), the quotient rule allows us to smoothly progress towards the derivative of the entire function.

Power Rule

The power rule is a fast track to derivatives for functions where our variable, usually \( x \), is raised to a power. It's like finding out how much more powerful a superhero becomes with each level up! In mathematical terms, if a function has the form \( f(x) = x^n \), where \( n \) is any real number, the power rule states that the derivative of \( f \) with respect to \( x \) is \( f'(x) = nx^{n-1} \). This rule simplifies the process of differentiation by reducing the exponent by one and multiplying by the original exponent.

For example, if we apply the power rule to the function \( u(x) = x^3 + 3x + 2 \) from our exercise, we will differentiate each term separately: \( u' = 3x^2 + 3 \). Notably, for the constant term, like the '+2' here, the derivative is zero because the power of \( x \) is zero, which aligns with the idea that constant terms do not change and thus have no rate of change.

Simplifying Derivatives

Simplifying derivatives is akin to cleaning up after baking a cake - it makes the result presentable and easier to understand. It involves combining like terms, factoring, and canceling out where possible to express the derivative in its simplest form. After using the quotient and power rules to find the raw derivative, in many cases—just like the given exercise—the result may appear complex and unwieldy at first glance.

When simplifying, look out for similar terms that can be combined, and remember that the objective is to minimize the expression without changing its value. For example, in our derivative \( \frac{2x^3 + 6x^2 + 6x - 2x^4 - 6x^3}{(x^2 + 1)^2} \), we assess terms across the numerator to see if any can be combined. In this case, negative and positive coefficients of \( x^3 \) terms can be added together to simplify the expression. Through simplification, understanding and working with the derivative becomes much more straightforward.