How do you differentiate y=(x+3) / root x?

Question

How do you differentiate y=(x+3) / root x

Answer 1

The easiest way is to do the division, and then use the power rule:

d((x+3)/√x)/dx
d(x^(1/2) + 3x^(-1/2))/dx
1/2 x^(-1/2) + 3/2 x^(-3/2)

And we are done.

Answer 2

I disagree. The easiest way is to bring the root x to the numerator. But, to each his own. Some will prefer the quotient rule.

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

Now, product rule.

[(x)^-(1/2)]+(-1/2)(x+3)*(x)^-(3/2)
(1/root(x))+(-1/2)(x+3)/(x^(3/2))

Answer 3

y = (x+3) / √x

quotien rule:
d/dx (u/v) = (u'v - v'u)/v^2
where u = x + 3
and v = √x

y' = [ d/dx (x+3) (√x) - d/dx (√x) (x+3)] / (√x)^2

y' = [ (1)(√x) - 1/(2√x) (x+3)] / x

y' = [ √x - (x+3)/(2√x)] / x

y' = [ (2x)/(2√x) - (x+3)/(2√x)] / x

y' = [ (2x - (x+3)) / (2√x)] / x

y' = [ (2x - x - 3) / (2√x)] / x

y' = [ (x-3) / (2√x)] / x

y' = (x-3) / [2x^(3/2)] <== answer