The problem asks to find the derivatives of two expressions: a) $x^4 - 3x^2 + 4x - 1$ b) $\frac{2x+3}{5x-1}$

1. Problem Description

The problem asks to find the derivatives of two expressions:

a) $x^4 - 3x^2 + 4x - 1$

b) $\frac{2x+3}{5x-1}$

2. Solution Steps

a) To find the derivative of $x^4 - 3x^2 + 4x - 1$, we use the power rule and the sum/difference rule.

The power rule states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

The sum/difference rule states that if $f(x) = u(x) \pm v(x)$, then $f'(x) = u'(x) \pm v'(x)$.

So,

$\frac{d}{dx}(x^4 - 3x^2 + 4x - 1) = \frac{d}{dx}(x^4) - \frac{d}{dx}(3x^2) + \frac{d}{dx}(4x) - \frac{d}{dx}(1)$

$= 4x^3 - 3(2x) + 4(1) - 0$

$= 4x^3 - 6x + 4$

b) To find the derivative of $\frac{2x+3}{5x-1}$, we use the quotient rule.

The quotient rule states that if $f(x) = \frac{u(x)}{v(x)}$, then $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$.

Here, $u(x) = 2x+3$ and $v(x) = 5x-1$.

Then, $u'(x) = 2$ and $v'(x) = 5$.

So,

$\frac{d}{dx}(\frac{2x+3}{5x-1}) = \frac{2(5x-1) - (2x+3)(5)}{(5x-1)^2}$

$= \frac{10x - 2 - (10x + 15)}{(5x-1)^2}$

$= \frac{10x - 2 - 10x - 15}{(5x-1)^2}$

$= \frac{-17}{(5x-1)^2}$

3. Final Answer

a) $4x^3 - 6x + 4$

b) $\frac{-17}{(5x-1)^2}$