a. \(y' = 32{\left( {x - {x^2}} \right)^{31}}\left( {1 - 2x} \right)\)
b. \(y' =- {{\left( {x\sqrt x } \right)'} \over {{x^3}}} = -{{\sqrt x + {x \over {2\sqrt x }}} \over {{x^3}}} = {{ - 3x} \over {2\sqrt x .{x^3}}} = {{ - 3} \over {2{x^2}\sqrt x }}\)
c. \(y' = {{\sqrt {1 - x} - \left( {1 + x} \right).{{ - 1} \over {2\sqrt {1 - x} }}} \over {1 - x}} = {{2\left( {1 - x} \right) + 1 + x} \over {2\sqrt {{{\left( {1 - x} \right)}^3}} }} = {{3 - x} \over {2\sqrt {{{\left( {1 - x} \right)}^3}} }}\)
d.
\(\eqalign{ & y' = {{\sqrt {{a^2} - {x^2}} - x.{{ - 2x} \over {2\sqrt {{a^2} - {x^2}} }}} \over {{{\left({\sqrt {{a^2} - {x^2}} } \right)}^2}}} = {{2\left( {{a^2} - {x^2}} \right) + 2{x^2}} \over {2{{\left( {\sqrt {{a^2} - {x^2}} } \right)}^3}}} \cr & = {{{a^2}} \over {\sqrt {{{\left( {{a^2} - {x^2}} \right)}^3}} }} \cr} \)
