Question

How do you differentiate `y=(x^2+4x+3)/sqrtx`?

Answer 1

`dy/dx=(3x^2+4x-3)/(2x^(3/2))`

Explanation:

What I'd prefer is to do is split up the fraction into its three parts by dividing each term by `sqrtx=x^(1/2)`:

`y=x^2/x^(1/2)+(4x)/x^(1/2)+3/x^(1/2)`

`y=x^(3/2)+4x^(1/2)+3x^(-1/2)`

Now instead of having to use the quotient rule, we can simply differentiate term-by-term using the power rule:

`dy/dx=3/2x^(1/2)+4*1/2x^(-1/2)+3*-1/2x^(-3/2)`

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

Finding a common denominator of `2x^(3/2)`:

`dy/dx=(3x^2+4x-3)/(2x^(3/2))`