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 = t2 + 2t
\(\rm {dy\over dt}\) = 2t + 2
Also x = t3
\(\rm dx\over dt\) = 3t2
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,
\(\rm {dy\over dx}\) = \(\rm \frac{2(5) +2}{3(5)^2}\) = \(\rm \frac{12}{75}\) = \(\rm \frac{4}{25}\)