Answer
$f'(x)=\dfrac{2(100x^{3}+40x^{2}+4x-1)}{(5x+1)^{2}}$
Work Step by Step
$f(x)=4x^{2}-\dfrac{2x}{5x+1}$
Evaluate the derivative term by term. Use the quotient rule to evaluate the derivative of the second term:
$f'(x)=(4x^{2})'-\Big(\dfrac{2x}{5x+1}\Big)'=...$
$...=8x-\dfrac{(5x+1)(2x)'-(2x)(5x+1)'}{(5x+1)^{2}}=...$
Evaluate the derivatives indicated and simplify:
$...=8x-\dfrac{2(5x+1)-5(2x)}{(5x+1)^{2}}=8x-\dfrac{10x+2-10x}{(5x+1)^{2}}=...$
$...=8x-\dfrac{2}{(5x+1)^{2}}=\dfrac{8x(5x+1)^{2}-2}{(5x+1)^{2}}=...$
$...=\dfrac{8x(25x^{2}+10x+1)-2}{(5x+1)^{2}}=\dfrac{200x^{3}+80x^{2}+8x-2}{(5x+1)^{2}}=...$
$...=\dfrac{2(100x^{3}+40x^{2}+4x-1)}{(5x+1)^{2}}$
