Question

How do you find the derivative of `x*x^(1/2)`?

Answer 1

`d/(dx) (x*x^(1/2)) = d/(dx) x^(3/2) = 3/2x^(1/2)`

or

`d/(dx) (x*x^(1/2)) = (d/(dx) x) * x^(1/2) + x * (d/(dx)x^(1/2))`

`=1*x^(1/2)+x*(1/2)x^(-1/2) = x^(1/2)+1/2 x^(1/2) = 3/2 x^(1/2)`

Explanation:

We can either multiply `x*x^(1/2) = x^(3/2)` first then use the power rule, or we can use the product rule, using the power rule on each part.

Power Rule
`d/(dx) x^k = k*x^(k-1)`

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