simple derivatives

Compute the derivatives of the following functions at points where the derivative exists:

Variant 1
\[ 5x^{2011} - x^3 - 2x + 1. \]
Result
\[ 10055x^{2010} - 3x^2 - 2. \] The derivative exists for all real \(x\).
Variant 2
\[ \frac{(2x-3)^{12}}{x^2 - x - 1}. \]
Result
\[ \frac{(2x - 3)^{11}(20x^2 - 16x - 27)}{(x^2 - x - 1)^2}. \] The derivative exists for \(x \in \Bbb{R} \setminus\left\{\frac{1 - \sqrt 5}2, \frac{1 + \sqrt 5}2 \right\}\).
Variant 3
\[ \frac{1 - \sqrt{5-2x}}{\sqrt{3x-8} + 2x} \]
Result
The function has an empty domain of definition, so the derivative does not exist at any point.
Variant 4
\[ \frac{\cos(\sqrt{x^2 + 4}) + e^{2x}}{\ln x}. \]
Result
\[ \frac {(2e^{2x} - \frac{x\sin(\sqrt{x^2 + 4})}{\sqrt{x^2+ 4}})\ln x - \frac{1}x\left(\cos(\sqrt{x^2 + 4} + e^{2x})\right)} {\ln^2 x}. \] The derivative exists for all \(x \in (0{,}1) \cup (1, \infty)\).