Question

What is the derivative of `xsqrt(1-x)`?

Answer 1

`= \frac{2-3x}{2\sqrt{1-x}}`

Explanation:

Question: `\frac{d}{dx} x(1-x)^{\frac{1}{2}}`

Use product rule, `\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + g'(x)f(x)`, power rule `\frac{d}{dx} [x^n] = nx^{n-1}` and chain rule, `\frac{d}{dx} [f'(g(x))] = f'(g(x)) * g'(x)`

`\frac{d}{dx} x(1-x)^{\frac{1}{2}} = 1(1-x)^{\frac{1}{2}} + \frac{d}{dx}[(1-x)^{\frac{1}{2}}] * x`

`= (1-x)^{\frac{1}{2}} + x * [\frac{1}{2}(1-x)^{\frac{-1}{2}} * -1]`

`= \sqrt{1-x} + x*[\frac{-1}{2} * \frac{1}{\sqrt{1-x}}]`

`= \sqrt{1-x} + \frac{-x}{2\sqrt{1-x}}`

`= \frac{2(1-x)}{2sqrt(1-x)} + \frac{-x}{2\sqrt{1-x}}`

`= \frac{2-3x}{2\sqrt{1-x}}`