Let's differentiate the function $g(x) = \dfrac{x-5}{x^2+1}$ using the quotient rule.

The quotient rule states that if $u(x)$ and $v(x)$ are differentiable functions of $x$, then:

$\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}$

Here, let: $u(x) = x - 5$ $v(x) = x^2 + 1$

First, we find the derivatives $u'(x)$ and $v'(x)$:

$u'(x) = \frac{d}{dx} (x - 5) = 1$

$v'(x) = \frac{d}{dx} (x^2 + 1) = 2x$

Now, we apply the quotient rule:

$g'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2} = \frac{(1)(x^2 + 1) - (x - 5)(2x)}{(x^2 + 1)^2}$

Next, we simplify the numerator:

$(1)(x^2 + 1) - (x - 5)(2x) = x^2 + 1 - 2x(x - 5) = x^2 + 1 - 2x^2 + 10x = -x^2 + 10x + 1$

So, we have:

$g'(x) = \frac{-x^2 + 10x + 1}{(x^2 + 1)^2}$

Therefore, the derivative of $g(x) = \dfrac{x-5}{x^2+1}$ is:

$g'(x) = \frac{-x^2 + 10x + 1}{(x^2 + 1)^2}$

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How do you apply the chain rule to composite functions?
2. What is the product rule, and how is it used in differentiation?
3. Can you explain how to find the critical points of a function?
4. How do you determine the concavity of a function using the second derivative?
5. What are the applications of the derivative in physics and engineering?

Tip: Always check your work for simplification errors, especially in complex differentiation problems. Simplifying expressions step-by-step helps to avoid mistakes and makes it easier to verify each part of the solution.