Correct Answer - Option 4 :

\(\rm \frac{4}{25}\)
__Concept:__

Parametric Form:

If f(x) and g(x) are the functions in x, then

\(\rm df(x)\over dg(x)\) = \(\rm \frac{df(x)\over dx}{dg(x)\over dx}\)

__Calculation:__

Given y = t^{2} + 2t

\(\rm {dy\over dt}\) = 2t + 2

Also x = t^{3}

\(\rm dx\over dt\) = 3t^{2}

Now \(\rm dy\over dx\) = \(\rm \frac{dy\over dt}{dx\over dt}\)

\(\rm dy\over dx\) = \(\rm \frac{2t +2}{3t^2}\)

At t = 5,

\(\rm {dy\over dx}\) = \(\rm \frac{2(5) +2}{3(5)^2}\) = \(\rm \frac{4}{25}\)