Differentiate this function using the chain rule?

Question

y = (x/(1-root(x)))^3

Answer 1

The function y = [x /(1 - √x)]³

Differentiating on each side with respect to x.

d/dx (y) = d/dx [x /(1 - √x)]³

              =3 [x /(1 - √x)]²  d/dx [x /(1 - √x)]

Apply quotient rule in derivatives.

d/dx[u/v] = (vu' - uv')/v²

In this case

u = x  

u' = 1

v = (1 - √x)

v' = d/dx (1 - √x)

v' = - 1/(2√x)

y' = 3 [x /(1 - √x)]^{2 *} {[(1 - √x) - (x)(- 1/(2√x))] /(1 - √x)²}

    =3x²(1 - √x)² [(1 - √x) + (√x/2)]/(1 - √x)⁶.

    = 3x²[(2 - √x)/2)]/(1 - √x)⁴.