N. Nog meer oefeningen
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}