I am a senior in high school so I know I am simply misunderstanding something but I don't know what, please have patience.
I was tasked to find the derivative for the following function:
$$ y = \frac{ (4x)^{1/5} }{5} + { \left( \frac{1}{x^3} \right) } ^ {1/4} $$
Simplifying:
$$ y = \frac{ 4^{1/5} }{5} x^{1/5} + { \frac{1 ^ {1/4}}{x ^ {3/4}} } $$
$$ y = \frac{ 4^{1/5} }{5} x^{1/5} + { \frac{\pm 1}{x ^ {3/4}} } $$
Because $ 1 ^ {1/n} = \pm 1 $, given $n$ is even
$$ y = \frac{ 4^{1/5} }{5} x^{1/5} \pm { x ^ {-3/4} } $$
Taking the derivative using power rule:
$$ \frac{dy}{dx} = \frac{ 4^{1/5} }{25} x^{-4/5} \pm \frac{-3}{4} { x ^ {-7/4} } $$
which is the same as
$$ \frac{dy}{dx} = \frac{ 4^{1/5} }{25} x^{-4/5} \pm \frac{3}{4} { x ^ {-7/4} } $$
And that is the part that I find difficult to understand. I know that I should be adding the second term(I graphed it multiple times to make sure), but I cannot catch my error and my teacher did't want to discuss it.
So I know I am doing something wrong because one function cannot have more than one derivative.