Given : `V=4/3pir^(3)`, find `(dV)/(dr)` and hence `(dV)/(dr)]_(r=2)`.

Answer

To solve the problem, we need to find the derivative of the volume \( V = \frac{4}{3} \pi r^3 \) with respect to \( r \), and then evaluate this derivative at \( r = 2 \).

### Step 1: Differentiate \( V \) with respect to \( r \)

Given:
\[ V = \frac{4}{3} \pi r^3 \]

To find \( \frac{dV}{dr} \), we apply the power rule of differentiation. The power rule states that if \( f(r) = r^n \), then \( \frac{df}{dr} = n r^{n-1} \).

Using this rule:
\[ \frac{dV}{dr} = \frac{d}{dr} \left( \frac{4}{3} \pi r^3 \right) \]

Since \( \frac{4}{3} \pi \) is a constant, we can factor it out:
\[ \frac{dV}{dr} = \frac{4}{3} \pi \cdot \frac{d}{dr}(r^3) \]

Now, differentiate \( r^3 \):
\[ \frac{d}{dr}(r^3) = 3r^2 \]

Substituting this back into our equation:
\[ \frac{dV}{dr} = \frac{4}{3} \pi \cdot 3r^2 \]

The \( 3 \) in the numerator and the \( 3 \) in the denominator cancel out:
\[ \frac{dV}{dr} = 4 \pi r^2 \]

### Step 2: Evaluate \( \frac{dV}{dr} \) at \( r = 2 \)

Now we need to find \( \frac{dV}{dr} \) when \( r = 2 \):
\[ \frac{dV}{dr} \bigg|_{r=2} = 4 \pi (2^2) \]

Calculating \( 2^2 \):
\[ 2^2 = 4 \]

Now substitute this back:
\[ \frac{dV}{dr} \bigg|_{r=2} = 4 \pi \cdot 4 = 16 \pi \]

### Final Answer
Thus, the value of \( \frac{dV}{dr} \) at \( r = 2 \) is:
\[ \frac{dV}{dr} \bigg|_{r=2} = 16 \pi \]

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