$\begin{array}{1 1} (A)\;\frac{27}{8}\pi(2x+1)^2 \\ (B)\;\frac{-27}{8}\pi(2x+1)^2 \\ (C)\;\frac{-27}{4}\pi(2x+1)^2 \\ (D)\;\frac{27}{4}\pi(2x+1)^2 \end{array} $

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- If $y=f(x)$,then $\large\frac{dy}{dx}$ measures the rate of change of $y$ w.r.t $x$.
- $\big(\large\frac{dy}{dx}\big)_{x=x_0}$ represents the rate of change of $y$ w.r.t $x$ at $x=x_0$

Step 1:

Diameter of the sphere =$\large\frac{3}{2}$$(2x+1)$

Therefore radius $r=\large\frac{3}{4}$$(2x+1)$

Volume of the sphere $v=\large\frac{4}{3}$$\pi r^3$

Substituting for $r$ we get,

$v=\large\frac{4}{3}$$\pi\big[\large\frac{3}{4}$$(2x+1)\big]^3$

Step 2:

Differentiating w.r.t $x$ on both sides,

$\large\frac{dv}{dx}=\frac{4}{3}$$x\times \big(\large\frac{3}{4}\big)^3$$\times 3(2x+1)^2(2)$

$\quad\;\;=\large\frac{27}{8}$$\pi(2x+1)^2$

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