Question

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

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

Answer 1

Answer: (D)

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