Differentiate y=(4x+x^-5)^1/3

This problem requires the use of chain rule since y is the
result of composition of 2 functions u = 4x+x^-5 and v =
u^(1/3).

We'll differentiate first the power, what's inside
brackets remaining unchanged, then we'll differentiate what's inside brackets, with
respect to x.

dy/dx = (1/3)*(4x+x^-5)^(1/3 -
1)*(4x+x^-5)'

dy/dx = (1/3)*(4x+x^-5)^(-2/3)*(4 +
-5*x^(-6))

dy/dx = (4 - 5/x^6)/3*(4x +
1/x^5)^(2/3)

dy/dx = [(4x^6 -
5)/x^6]/3*[(4x^6+1)/x^5]^(2/3)

The result of
differentiating y is:

dy/dx = [(4x^6 -
5)/x^6]/3*[(4x^6+1)/x^5]^(2/3)