Question

How do you differentiate `f(t)=(2t)/(2+sqrtt)`?

Answer 1

Using the quotient rule allows for this to be differentiated.

Explanation:

Quotient rule:
`f(x) = g(x)/(h(x))`
`f'(x) = (g'(x)h(x) - h'(x)g(x))/(h(x)^2)`

In your question:
`f(t) = (2t)/(2 + sqrt(t))`

`g(t) = 2t`
`h(t) = 2 + sqrt(t)`

`g'(t) = 2`
`h'(t) = 1/sqrt(t)`

This means that
`f'(t) = (2(2 + sqrt(t))- 1/sqrt(t)2t)/((2+sqrt(t))^2)`
`f'(t) = (4+ 2sqrtt - 2sqrtt)/((2+sqrt(t))^2)`

Final answer:
`f'(t) = 4/((2+ sqrtt)^2)`