Question

How do you use the Product Rule to find the derivative of `y = (x^2 - 1) sqrtx`?

Answer 1

`color(red)(dy/dx= (5x^2-1)/(2sqrtx))`

Explanation:

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

The Product Rule, states that if `y=uv`, then

`dy/dx = u(dv)/dx+v(du)/dx`

Let `u=x^2-1` and `v=x^(1/2)`

Then

`(du)/dx=2x`

`(dv)/dx=1/2x^(-1/2)`

`dy/dx = u(dv)/dx+v(du)/dx`

`dy/dx=(x^2-1)× 1/2x^(-1/2)+ x^(1/2)×2x`

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

`dy/dx= (x^2-1)/(2x^(1/2)) +(4x^2)/(2x^(1/2))`

`dy/dx= (5x^2-1)/(2sqrtx)`