I believe the answer is
the absolute value function, $y=|x|$.
I am within a secret close to my writer,
The answer can be found in something that belongs to TheGreatDuck, for example, a document on his profile.
Implication is the key,
The document in question concerns work on implied calculus.
Within this document one can find
some purple text which reads: TEXT, TWICE, DERIVATIVE, SQUARE, ABSOLUTE.
Presumably these complement the lines of the riddle, but they are not necessarily in the right order.
Go forth to thee and once fifth,
This tells us in which order to plug the words into the unfinished lines: the first word to the forth line, the second word to the second line, the third word to the third line, the fourth word to the first line, and the fifth word to the fifth line. (Thanks to greenturtle3141 for this observation.)
I am the root of a...
square, since the root of $x^2$ is $\pm x$.
My sides are...
twice, since there are two sides to this equation, one on each side of the y-axis.
I have no...
derivative, since the function $y=|x|$ has no derivative at $x=0$.
But in purple...
My equation is...
absolute, since the equation $y=|x|$ involves an absolute.