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Alenkasestr [34]
3 years ago
8

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

Mathematics
1 answer:
aliina [53]3 years ago
6 0

Answer:  see proof below

<u>Step-by-step explanation:</u>

Use the Quotient rule for derivatives:

\text{If}\ y=\dfrac{a}{b}\quad \text{then}\ y'=\dfrac{a'b-ab'}{b^2}

Given: y=\dfrac{2\sqrtx}{1-x}

\sqrtxa=2\sqrt x\qquad  \rightarrow \qquad a'=\dfrac{1}{\sqrt x}\\\\b=1-x\qquad \rightarrow \qquad b'=-1        

y'=\dfrac{\dfrac{1-x}{\sqrt x}-(-2\sqrt x)}{(1-x)^2}\\\\\\.\quad =\dfrac{\dfrac{1-x}{\sqrt x}-(-2\sqrt x)\bigg(\dfrac{\sqrt x}{\sqrt x}\bigg)}{(1-x)^2}\\\\\\.\quad =\dfrac{1-x+2x}{\sqrt x(1-x)^2}\\\\\\.\quad =\dfrac{x+1}{\sqrt x(1-x)^2}

LHS = RHS:  \dfrac{x+1}{\sqrt x(1-x)^2}=\dfrac{x+1}{\sqrt x(1-x)^2}\qquad \checkmark

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