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

##### 2 Answers

#### Explanation:

The expression is product of two functions in

Denoting these by

the first is

and the second is

The derivative of the expression is

The derivative of the first function is straightforward

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

writing the compound function as

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

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

And the derivative of the inner function is

So the derivative of the compound function

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

So the overall derivative is

#### Explanation:

and,