The area of a circle is given by `A= pi r^(2)`, where r is the radius. Calculate the rate of increase of area w.r.t radius.

Explanation

To solve the problem of calculating the rate of increase of the area of a circle with respect to its radius, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Area**:
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.

2. **Differentiate the Area with Respect to Radius**:
To find the rate of increase of the area with respect to the radius, we need to differentiate the area \( A \) with respect to \( r \). This is represented as:
\[ \frac{dA}{dr} \]

3. **Apply the Derivative**:
Using the power rule of differentiation, we differentiate \( A = \pi r^2 \):
\[ \frac{dA}{dr} = \pi \cdot \frac{d}{dr}(r^2) \]
The derivative of \( r^2 \) is \( 2r \), so we have:
\[ \frac{dA}{dr} = \pi \cdot 2r \]

4. **Simplify the Expression**:
Simplifying the expression gives:
\[ \frac{dA}{dr} = 2\pi r \]

5. **Final Result**:
Therefore, the rate of increase of the area with respect to the radius is:
\[ \frac{dA}{dr} = 2\pi r \]

### Final Answer:
The rate of increase of the area of a circle with respect to its radius is \( 2\pi r \).

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