y = e3x x3 \(\frac{dy}{dx}\) at x = \(\frac{1}{3}\)
\(\frac{dy}{dx}\) = \(\left[x^3 . \frac{d}{dx}^{(e^{3x})} + e^{3x} . \frac{d}{dx}^{(x^3)}\right]\) [By product rule]
= x3 . e3x . 3 + e3x . 3x2 [By chain rule]
= 3e3x x3 + e3x . 3x2
= 3x2 [e3x . x + e3x]
Put x = \(\frac{1}{3}\)
= 3 (\(\frac{1}{3}\))2 \(\left[e^{3{(1/3)}}.\frac{1}{3} + e^{3{(1/3)}}\right]\)
= 3 × \(\frac{1}{9}\)[\(\frac{e}{3}\) + e]
= \(\frac{1}{3}\)\(\left[\frac{e + 3e}{3}\right]\)
= \(\frac{4e}{9}\)