Question

Find dy / dx for the following functions :-
y = $\sf{\frac{(x-1)(x-2)}{\sqrt{x}}}$

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Answer 1

Hope it helps u pls check it out

Answer 2

Answer:

$( \frac{dy}{dx} ) = ( \frac{(x - 1)(x - 2)}{ \sqrt{x} } )( \frac{1}{x - 1} + \frac{1}{x - 2} - \frac{1}{2x} )$

Step-by-step explanation:

I'm assuming the function is

$y = \frac{(x - 1)(x - 2}{ \sqrt{x} }$

I will use logarithmic differentiation to compute this derivative. If we take the natural logarithm of both sides, we get:

$ln \: y = ln( \frac{(x - 1)(x - 2)}{ \sqrt{x} }$

Using the laws of logarithms to simplify:

$ln \: y = ln(x - 1) + ln(x - 2) - ln \sqrt{x} \\ ln \: y = ln(x - 1) + ln(x - 2) - ln( {x}^{ \frac{1}{2} } ) \\ ln \: y = ln(x - 1) + ln(x - 2) - \frac{1}{2} ln(x)$

$\frac{1}{y} ( \frac{dy}{dx} ) = \frac{1}{x - 1} + \frac{1}{x - 2} - \frac{1}{2x} \\ ( \frac{dy}{dx} ) = y( \frac{1}{x - 1} + \frac{1}{x - 2} - \frac{1}{2x} ) \\ ( \frac{dy}{dx} ) = ( \frac{(x - 1)(x - 2)}{ \sqrt{x} } )( \frac{1}{x - 1} + \frac{1}{x - 2} - \frac{1}{2x} )$

Hopefully this helps!