Let's solve the derivatives step by step:

a. $y(x) = 6x^2 - 10x - 5x^{-2}$

Take the derivative term by term: $\frac{dy}{dx} = \frac{d}{dx}(6x^2) - \frac{d}{dx}(10x) - \frac{d}{dx}(5x^{-2})$ $\frac{dy}{dx} = 12x - 10 + 10x^{-3}$

Result for (a):
$\boxed{\frac{dy}{dx} = 12x - 10 + \frac{10}{x^3}}$

b. $r(s) = \frac{1}{s^3} - \frac{5}{2s}$

Differentiate each term: $\frac{dr}{ds} = \frac{d}{ds}(s^{-3}) - \frac{d}{ds}\left(\frac{5}{2}s^{-1}\right)$ $\frac{dr}{ds} = -3s^{-4} + \frac{5}{2}s^{-2}$

Result for (b):
$\boxed{\frac{dr}{ds} = -\frac{3}{s^4} + \frac{5}{2s^2}}$

c. $y(x) = (3 - x^2)(x^3 - x + 1)$

Use the product rule: $(uv)' = u'v + uv'$, where $u = (3 - x^2)$ and $v = (x^3 - x + 1)$.
First, compute derivatives of $u$ and $v$:
$u' = \frac{d}{dx}(3 - x^2) = -2x, \quad v' = \frac{d}{dx}(x^3 - x + 1) = 3x^2 - 1$ Apply the product rule: $\frac{dy}{dx} = u'v + uv'$ $\frac{dy}{dx} = (-2x)(x^3 - x + 1) + (3 - x^2)(3x^2 - 1)$ Simplify: $\frac{dy}{dx} = -2x^4 + 2x^2 - 2x + 9x^2 - 3x^4 - 3x^2 + x^2$ $\frac{dy}{dx} = -5x^4 + 9x^2 - 2x$

Result for (c):
$\boxed{\frac{dy}{dx} = -5x^4 + 9x^2 - 2x}$

d. $z(x) = \frac{4 - 3x}{3x^2 + x}$

Use the quotient rule:
$\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$ Here, $f = 4 - 3x$ and $g = 3x^2 + x$.
Compute derivatives of $f$ and $g$:
$f' = \frac{d}{dx}(4 - 3x) = -3, \quad g' = \frac{d}{dx}(3x^2 + x) = 6x + 1$ Apply the quotient rule: $\frac{dz}{dx} = \frac{(-3)(3x^2 + x) - (4 - 3x)(6x + 1)}{(3x^2 + x)^2}$ Simplify the numerator: $(-3)(3x^2 + x) = -9x^2 - 3x$ $(4 - 3x)(6x + 1) = 24x + 4 - 18x^2 - 3x = -18x^2 + 21x + 4$ $\text{Numerator: } -9x^2 - 3x - (-18x^2 + 21x + 4) = 9x^2 - 24x - 4$ Thus: $\frac{dz}{dx} = \frac{9x^2 - 24x - 4}{(3x^2 + x)^2}$

Result for (d):
$\boxed{\frac{dz}{dx} = \frac{9x^2 - 24x - 4}{(3x^2 + x)^2}}$

Let me know if you want detailed steps for any specific derivative!

Related Questions:

1. How does the product rule apply when three terms are multiplied?
2. What is the general rule for differentiating a quotient?
3. How do we handle derivatives involving fractional exponents?
4. How can implicit differentiation be applied to these types of problems?
5. Can higher-order derivatives be computed for these functions?

Tip:

Always simplify each step in differentiation carefully to avoid errors, especially when using the product and quotient rules.