We need to differentiate:

$y = x + \frac{5}{x^3 + x - 1}$

Step 1: Differentiate the First Term

The derivative of $x$ is simply:

$\frac{d}{dx} (x) = 1$

Step 2: Differentiate the Second Term Using the Quotient Rule

The second term is a fraction, so we apply the quotient rule, which states:

$\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x) g(x) - f(x) g'(x)}{[g(x)]^2}$

where:

• $f(x) = 5 \Rightarrow f'(x) = 0$
• $g(x) = x^3 + x - 1 \Rightarrow g'(x) = 3x^2 + 1$

Applying the quotient rule:

\frac{0 \cdot (x^3 + x - 1) - 5(3x^2 + 1)}{(x^3 + x - 1)^2}$$ \[ = \frac{-5(3x^2 + 1)}{(x^3 + x -