1.
$
\eqalign{
& f(x) = \sqrt {\frac{x}
{{x - 1}}} \cr
& f'(x) = \frac{1}
{{2 \cdot \sqrt {\frac{x}
{{x - 1}}} }} \cdot \frac{{ - 1}}
{{\left( {x - 1} \right)^2 }} \cr
& f'(x) = \frac{1}
{{2 \cdot \sqrt {\frac{x}
{{x - 1}}} }} \cdot \frac{{ - 1}}
{{\left( {x - 1} \right)^2 }} \cdot \frac{{\sqrt {\frac{x}
{{x - 1}}} }}
{{\sqrt {\frac{x}
{{x - 1}}} }} \cr
& f'(x) = \frac{1}
{{2 \cdot \frac{x}
{{x - 1}}}} \cdot \frac{{ - 1\sqrt {\frac{x}
{{x - 1}}} }}
{{\left( {x - 1} \right)^2 }} \cr
& f'(x) = \frac{1}
{{2x}} \cdot \frac{{ - 1\sqrt {\frac{x}
{{x - 1}}} }}
{{x - 1}} \cr
& f'(x) = - \frac{{\sqrt {\frac{x}
{{x - 1}}} }}
{{2x\left( {x - 1} \right)}} \cr}
$
2.
$
\eqalign{
& f(x) = 1 + \sqrt {x^2 - x} \cr
& f'(x) = \frac{1}
{{2\sqrt {x^2 - x} }} \cdot \left( {2x - 1} \right) = \frac{{2x - 1}}
{{2\sqrt {x^2 - x} }} \cr
& f''(x) = \frac{{2 \cdot 2\sqrt {x^2 - x} - \left( {2x - 1} \right) \cdot \frac{{2x - 1}}
{{\sqrt {x^2 - x} }}}}
{{\left( {2\sqrt {x^2 - x} } \right)^2 }} \cr
& f''(x) = \frac{{4\sqrt {x^2 - x} - \frac{{\left( {2x - 1} \right)^2 }}
{{\sqrt {x^2 - x} }}}}
{{\left( {2\sqrt {x^2 - x} } \right)^2 }} \cr
& f''(x) = \frac{{\frac{{4\sqrt {x^2 - x} \cdot \sqrt {x^2 - x} }}
{{\sqrt {x^2 - x} }} - \frac{{\left( {2x - 1} \right)^2 }}
{{\sqrt {x^2 - x} }}}}
{{4\left( {\sqrt {x^2 - x} } \right)^2 }} \cr
& f''(x) = \frac{{\frac{{4\left( {x^2 - x} \right) - \left( {2x - 1} \right)^2 }}
{{\sqrt {x^2 - x} }}}}
{{4\left( {\sqrt {x^2 - x} } \right)^2 }} \cr
& f''(x) = \frac{{4\left( {x^2 - x} \right) - \left( {2x - 1} \right)^2 }}
{{4\left( {\sqrt {x^2 - x} } \right)^3 }} \cr
& f''(x) = \frac{{ - 1}}
{{4\left( {\sqrt {x^2 - x} } \right)^3 }} \cr
& f''(x) = - \frac{1}
{{4\left( {\sqrt {x^2 - x} } \right)^3 }} \cr}
$
3.
$
f(x) = \large\frac{4}{{\sqrt {2x^3 + 5x^2 + 4} }}
$
$
f(x) = 4\left( {2x^3 + 5x^2 + 4} \right)^{ - \frac{1}{2}}
$
Dan de afgeleide:
$
f'(x) = 4 \cdot - \frac{1}{2}\left( {2x^3 + 5x^2 + 4} \right)^{ - 1\frac{1}{2}} \cdot \left( {6x^2 + 10x} \right)
$
$
f'(x) = - \large\frac{{2\left( {6x^2 + 10x} \right)}}{{\sqrt {\left( {2x^3 + 5x^2 + 4} \right)^3 } }}
$
$
f'(x) = - \large\frac{{12x^2 + 20x}}{{\sqrt {\left( {2x^3 + 5x^2 + 4} \right)^3 } }}
$
