6. Find (dy)/(dx) if:
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems.
6. Find $\frac{dy}{dx}$ if:
a) $y = \frac{x^2 - 1}{x^2 + 1}$
Step 1: Use the quotient rule, $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}$.
Let $u = x^2 - 1$, so $u' = 2x$.
Let $v = x^2 + 1$, so $v' = 2x$.
Step 2: Substitute $u, u', v, v'$ into the quotient rule formula.
$$ \frac{dy}{dx} = \frac{(
