Simplifying Derivatives

example 1

$a'(x)=x^{3}{(2x-5)}^{4}$

Product Rule $(f*g)'=f*g'+g*f'$

$f=x^{3}$ $f'=3x^{2}$ $g={(2x-5)}^{4}$ $g'=4{(2x-5)}^{3}(2)$

$a'(x)=(3x^{2})({(2x-5)}^{4})+(x^{3})(4{(2x-5)}^{3}(2))$

$a'(x)=(3x^{2})({(2x-5)}^{4})+(x^{3})(8{(2x-5)}^{3})$

factorize

$a'(x)=x^{2}({(2x-5)}^{3})(8x)+3(2x-5))$

$a'(x)=x^{2}({(2x-5)}^{3})(8x+6x-15))$

$a'(x)=x^{2}{(2x-5)}^{3}(14x-15))$

example 2

$f'(x)=x^{2}{\sqrt[{2}]{4-9x}}$ simplify

Chain rule $[f(g(x))]'=f'(g(x))*g'(x)$

$2x{\sqrt[{2}]{4-9x}}+{-9 \over 2}{(4-9x)}^{-1 \over 2}$

${x(8-{45x \over 2}){\sqrt[{2}]{4-9x}} \over {\sqrt[{2}]{4-9x}}}$