Question

How do you find the derivative of `xsqrt (1-x)`?

Answer 1

`(2- 3x)/(2 sqrt(1-x))`

Explanation:

The expression is product of two functions in `x`.

Denoting these by `f(x)` and `g(x)`, respectively,

the first is
`f(x) = x`

and the second is
`g(x) = sqrt(1 - x)`

`g(x)` is a compound function (ie ie a function of a function)

The derivative of the expression is

`f'(x)g(x) + f(x)g'(x)`

The derivative of the first function is straightforward
`f'(x) = 1`

The derivative of the second is trickier because it is a compound function. This requires the chain rule. The outer function is the square root function, and the inner function is the polynomial `(1 - x)`

writing the compound function as
`h(j(x))` (`h` of `j` of `x`), the derivative is

`h'(j(x))j'(x)`

That is, the derivative of the outer function evaluated at the inner function times the derivative of the inner function

It makes things simpler to rewrite `g(x)` using index notation, that is

`g(x) = (1 - x)^(1/2)`

Evaluating the outer function is the straightforward application of the rules of polynomial differentiation applied to its index, that is

`h'(j(x)) = 1/2(1-x)^(1/2 - 1) = 1/2(1-x)^(-1/2)`

And the derivative of the inner function is
`j'(x) = -1`

So the derivative of the compound function `g(x)` is

`g'(x) = h'(j(x))j'(x) = 1/2(1-x)^(-1/2)(-1)`
`= -1/2 (1 - x )^(-1/2)`

Or, if you prefer, reverting to the square root notation and noting the negative index

`g'(x) = - 1/(2 sqrt(1-x))`

So the overall derivative is

`f'(x)g(x) + f(x)g'(x)`

`= (1)(1 - x)^(1/2)+ (x)(-1/2 (1 - x )^(-1/2))`

`= sqrt(1-x) - x/(2 sqrt(1-x))`

`= (2(1-x))/(2 sqrt(1-x)) - x/(2 sqrt(1-x))`

`= (2- 2x- x)/(2 sqrt(1-x))`

`= (2- 3x)/(2 sqrt(1-x))`

Answer 2

`(2-3x)/(2sqrt(1-x)`

Explanation:

`color(red)(d/(dx)(u*v)=u*(dv)/(dx)+v*(du)/(dx))`
and,`d/(dx)(sqrt(1-x))=d/(dx)(1-x)^(1/2)=1/2*(1-x)^(-1/2)=1/2*1/sqrt(1-x)`
`y=x*sqrt(1-x)`
`=>(dy)/(dx)=x*d/(dx)(sqrt(1-x))+sqrt(1-x)*d/(dx)(x)`
`=>(dy)/(dx)=x*1/(2sqrt(1-x))(-1)+sqrt(1-x)*1`
`=(-x)/(2sqrt(1-x))+sqrt(1-x)=(-x+2(1-x))/(2sqrt(1-x)`
`=(-x+2-2x)/(2sqrt(1-x))=(2-3x)/(2sqrt(1-x)`