Answer
$f'(a)=\frac{1}{(1-4a)^2}$
Work Step by Step
$f'(a)=\lim\limits_{x \to a}\frac{f(x)-f(a)}{x-a}$ (Use the function formula $f(x)=\frac{x}{1-4x}$)
$=\lim\limits_{x \to a}\frac{\frac{x}{1-4x}-\frac{a}{1-4a}}{x-a}$ (Simplify the numerator)
$=\lim\limits_{x \to a}\frac{\frac{x(1-4a)-a(1-4x)}{(1-4x)(1-4a)}}{x-a}$
$=\lim\limits_{x \to a}\frac{\frac{x-a}{(1-4x)(1-4a)}}{x-a}$ (Cancel the common factor)
$=\lim\limits_{x \to a}\frac{1}{(1-4x)(1-4a)}$ (Evaluate the limit by direct substitution)
$=\frac{1}{(1-4a)(1-4a)}$
$=\frac{1}{(1-4a)^2}$
