Question

How do you differentiate `f(x)=xsinsqrtx` using the chain rule?

Answer 1

`sinsqrtx + (cossqrtx * sqrtx)/2`

Explanation:

In order to differentiate `f(x) = xsinsqrtx`, we need to use the product rule

Product rule: `f(x)*g(x) = f'(x)g(x) + g'(x)f(x)`

`f(x)*g(x) = f'(x)g(x) + g'(x)f(x)`
`x*sinsqrtx = d/dx x * sinsqrtx + d/dx sinsqrtx * x`

`x' = 1`
`sinsqrtx' = cossqrtx * d/dx sqrtx`

`1 * sinsqrtx + cossqrtx * d/dx sqrtx * x`
`sqrtx' = x^(1/2)' = (1/2)*x^((1/2) -1)= 1/(2sqrtx)`

`sinsqrtx + cossqrtx * 1/(2sqrtx) * x`
`sinsqrtx + (cossqrtx * sqrtx)/2`