Opgave 1
$
\eqalign{
& y = \frac{{3 - 2x}}
{{3 + 2x}} \cr
& y' = \frac{{ - 2 \cdot (3 + 2x) - (3 - 2x) \cdot 2}}
{{\left( {3 + 2x} \right)^2 }} \cr
& y' = \frac{{ - 6 - 4x - (6 - 4x)}}
{{\left( {3 + 2x} \right)^2 }} \cr
& y' = \frac{{ - 6 - 4x - 6 + 4x}}
{{\left( {3 + 2x} \right)^2 }} \cr
& y' = \frac{{ - 12}}
{{\left( {3 + 2x} \right)^2 }} \cr}
$
Opgave 2
$
\eqalign{
& y = \frac{{x^2 }}
{{\sqrt {4 - x^2 } }} \cr
& y' = \frac{{2x \cdot \sqrt {4 - x^2 } - x^2 \cdot \frac{1}
{{2\sqrt {4 - x^2 } }} \cdot - 2x}}
{{\left( {\sqrt {4 - x^2 } } \right)^2 }} \cr
& y' = \frac{{2x \cdot \sqrt {4 - x^2 } + 2x^3 \cdot \frac{1}
{{2\sqrt {4 - x^2 } }}}}
{{4 - x^2 }} \cr
& y' = \frac{{2x \cdot \left( {4 - x^2 } \right) + x^3 }}
{{\left( {4 - x^2 } \right) \cdot \sqrt {4 - x^2 } }} \cr
& y' = \frac{{8x - 2x^3 + x^3 }}
{{\left( {4 - x^2 } \right) \cdot \sqrt {4 - x^2 } }} \cr
& y' = \frac{{8x - x^3 }}
{{\left( {4 - x^2 } \right) \cdot \sqrt {4 - x^2 } }} \cr}
$
Opgave 3
$
\eqalign{
& y = x^5 + 5x^4 - 10x^2 + 6 \cr
& y' = 5x^4 + 20x^3 - 20x \cr
& y' = 5x\left( {x^3 + 4x^2 - 4} \right) \cr}
$
Opgave 4
$
\eqalign{
& y = 3x^{\frac{1}
{2}} - x^{\frac{3}
{2}} + 2x^{ - \frac{1}
{2}} \cr
& y' = \frac{3}
{2}x^{ - \frac{1}
{2}} - \frac{3}
{2}x^{\frac{1}
{2}} - x^{ - 1\frac{1}
{2}} \cr}
$
Opgave 5
$
\eqalign{
& y = \sqrt {2x} + 2\sqrt x \cr
& y' = \frac{1}
{{2\sqrt {2x} }} \cdot 2 + 2 \cdot \frac{1}
{{2\sqrt x }} \cr
& y' = \frac{1}
{{\sqrt {2x} }} + \frac{1}
{{\sqrt x }} \cr}
$
Opgave 6
$
\eqalign{
& y = \sqrt {3 + 4x - x^2 } \cr
& y' = \frac{1}
{{2\sqrt {3 + 4x - x^2 } }} \cdot \left( {4 - 2x} \right) \cr
& y' = \frac{{2 - x}}
{{\sqrt {3 + 4x - x^2 } }} \cr
& y' = \frac{{2 - x}}
{{\sqrt {3 + 4x - x^2 } }} \cr}
$