To find the derivative of a fraction, you can use the quotient rule. The quotient rule states that if you have a function of the form f(x) = g(x) / h(x), where both g(x) and h(x) are functions, then the derivative of f(x) with respect to x is given by:

f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2

Here, g'(x) represents the derivative of g(x) with respect to x, and h'(x) represents the derivative of h(x) with respect to x.

Let’s go through an example to illustrate this. Suppose you have the function f(x) = (3x^2 + 2x + 1) / (x^2 + 1). To find the derivative f'(x), we can apply the quotient rule as follows:

f'(x) = [(2 * 3x^2 + 2 * 1) * (x^2 + 1) – (3x^2 + 2x + 1) * (2x)] / (x^2 + 1)^2

Simplifying this expression will yield the derivative of the fraction f(x) with respect to x.

Expanding the numerator:

f'(x) = (6x^2 + 2) * (x^2 + 1) – (6x^4 + 4x^3 + 2x^2 + 2x) / (x^2 + 1)^2

Now, simplifying further:

f'(x) = 6x^4 + 6x^2 + 2 – 6x^4 – 4x^3 – 2x^2 – 2x / (x^2 + 1)^2

Combining like terms:

f'(x) = -4x^3 + 4x^2 – 2x / (x^2 + 1)^2

Thus, the derivative of the given fraction f(x) = (3x^2 + 2x + 1) / (x^2 + 1) is f'(x) = -4x^3 + 4x^2 – 2x / (x^2 + 1)^2.

Here are two more examples of finding the derivatives of fractions using the quotient rule:

Example 1: Let’s find the derivative of the fraction f(x) = (2x^3 – 3x) / (x^2 + 1).

Using the quotient rule, we have:

f'(x) = [(3 * 2x^2 – 3) * (x^2 + 1) – (2x^3 – 3x) * (2x)] / (x^2 + 1)^2

Simplifying:

f'(x) = (6x^2 – 3) * (x^2 + 1) – (4x^4 – 6x^2) / (x^2 + 1)^2

Expanding and combining like terms:

f'(x) = 6x^4 + 6x^2 – 3x^2 – 3 – 4x^4 + 6x^2 / (x^2 + 1)^2

Simplifying further:

f'(x) = 2x^4 + 9x^2 – 3 / (x^2 + 1)^2

So, the derivative of f(x) = (2x^3 – 3x) / (x^2 + 1) is f'(x) = 2x^4 + 9x^2 – 3 / (x^2 + 1)^2.

Example 2: Let’s find the derivative of the fraction g(x) = (5x + 1) / x^3.

Using the quotient rule:

g'(x) = [(1 * x^3) – (5x + 1) * (3x^2)] / (x^3)^2

Simplifying:

g'(x) = x^3 – (15x^3 + 3x^2) / x^6

Combining like terms:

g'(x) = -14x^3 – 3x^2 / x^6

Simplifying further:

g'(x) = -14/x^3 – 3/x^4

Therefore, the derivative of g(x) = (5x + 1) / x^3 is g'(x) = -14/x^3 – 3/x^4.