Question

If $y=\frac{1}{4}{u}^4$ and $u=\frac{2}{3}{x}^3+5$, then $\frac{dy}{dx}$ is equal to

• A. $x^2$
• B. $2x^3+15$
• C. $x^2+10$
• D. $\frac{2}{27}{x}^2{\left(2x^3+15\right)}^3$
• E. None of these

Answer 1

Answer: (D)

We have, $y=\frac{1}{4}{u}^4$
$\therefore \frac{dy}{du}=\frac{1}{4}4u^3=u^3$
also $u=\frac{2}{3}{x}^3+5$
$\therefore \frac{du}{dx}=\frac{2}{3}\cdot 3x^2=2x^2$
$\therefore \frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}=u^3\times 2x^2={\left(\frac{2}{3}{x}^3+5\right)}^3\left(2x^2\right)$
$=\frac{2}{27}{x}^2{\left(2x^3+15\right)}^3$