Question

How to find the derivative of the below question?
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Answer 1

Given:

`y=(x)/(√(x-1))`

To find:

The derivative of the given problem.

Solution:

Chain rule :
`(d)/(dx)f(g(x))=f'(g(x))\cdot g'(x)`

Quotient rule :
`(d)/(dx)\left((f(x))/(g(x))\right)=(g(x)f'(x)-f(x)g'(x))/([g(x)]^2)`

We have,

`y=(x)/(√(x-1))`

Differentiating with respect to x using chain rule and quotient rule, we get

`(dy)/(dx)=(√(x-1)(d)/(dx)(x)-x(d)/(dx)(√(x-1)))/((√(x-1))^2)`

`(dy)/(dx)=(√(x-1)(1)-x(1)/(2√(x-1))(d)/(dx)(x-1))/((x-1))`

`(dy)/(dx)=(√(x-1)-(x)/(2√(x-1))(1-0))/(x-1)`

`(dy)/(dx)=(√(x-1)-(x)/(2√(x-1)))/(x-1)`

`(dy)/(dx)=\frac{\frac{2x-2-x}{2(x-1)^{(1)/(2)}}}{(x-1)}`

`(dy)/(dx)=\frac{x-2}{2(x-1)^{(3)/(2)}}`

Therefore, the required derivative is
`(dy)/(dx)=\frac{x-2}{2(x-1)^{(3)/(2)}}`.