Find dy / dx for the following functions :-
y = [tex]\sf{\frac{(x-1)(x-2)}{\sqrt{x}}}[/tex]
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Answer:
[tex]( \frac{dy}{dx} ) = ( \frac{(x - 1)(x - 2)}{ \sqrt{x} } )( \frac{1}{x - 1} + \frac{1}{x - 2} - \frac{1}{2x} )[/tex]
Step-by-step explanation:
I'm assuming the function is
[tex]y = \frac{(x - 1)(x - 2}{ \sqrt{x} } [/tex]
I will use logarithmic differentiation to compute this derivative. If we take the natural logarithm of both sides, we get:
[tex]ln \: y = ln( \frac{(x - 1)(x - 2)}{ \sqrt{x} } [/tex]
Using the laws of logarithms to simplify:
[tex]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)[/tex]
[tex] \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} )[/tex]
Hopefully this helps!