Question

If y = 2√x÷ 1–x', show that dy÷dx = x+1 ÷ √x(1–x)²​

Answer 1

Answer: see proof below

Explanation:

Use the Quotient rule for derivatives:

`\text{If}\ y=(a)/(b)\quad \text{then}\ y'=(a'b-ab')/(b^2)`

Given:
`y=(2\sqrtx)/(1-x)`

`\sqrtx`
`a=2\sqrt x\qquad \rightarrow \qquad a'=(1)/(\sqrt x)\\\\b=1-x\qquad \rightarrow \qquad b'=-1`

`y'=((1-x)/(\sqrt x)-(-2\sqrt x))/((1-x)^2)\\\\\\.\quad =((1-x)/(\sqrt x)-(-2\sqrt x)\bigg((\sqrt x)/(\sqrt x)\bigg))/((1-x)^2)\\\\\\.\quad =(1-x+2x)/(\sqrt x(1-x)^2)\\\\\\.\quad =(x+1)/(\sqrt x(1-x)^2)`

LHS = RHS:
`(x+1)/(\sqrt x(1-x)^2)=(x+1)/(\sqrt x(1-x)^2)\qquad \checkmark`