Question

If $y=\frac{u^4}{4}andu=\frac{2}{3}{x}^3+5then\frac{dy}{dx}=$ ?

• A. $\frac{1}{27}{x}^2{({2x}^3+15)}^3$
• B. $\frac{2}{27}x{({2x}^3+5)}^3$
• C. $\frac{2}{27}{x}^2{({2x}^3+15)}^3$
• D. $noneofthese$

Answer 1

The correct option is C $\frac{2}{27}{x}^2{({2x}^3+15)}^3$

Given,

$y=\frac{u^4}{4}$ and $u=\frac{2}{3}{x}^3+5$.

Now differentiating both sides with respect to $x$ we get,

$\frac{dy}{du}=u^3$.....(1) and $\frac{du}{dx}=2x^2$.....(2).

Now,

$\frac{dy}{dx}$

$=\frac{\frac{dy}{du}}{\frac{dx}{du}}$

$=\frac{\frac{1}{3^3}(2x^3+15{)}^3}{\frac{1}{2x^2}}$ [ Using (1) and (2)]

$=\frac{2x^2}{27}(2x^3+15{)}^3$.