(a).

The formula for area  of the circle is .

The formula for the volume  of the sphere is .

.

Differentiate with respect to .

.

.

Differentiate with respect to .

.

(b).

The derivative of the formula for the area of the circle is the formula for the circumference

of the circle.

The derivative for the volume of the sphere is the formula for the surface area of the sphere.

(c).

A apothem of a regular polygon is a line segment from the center to the midpoint of one of its sides.

Figure:

Figure (i) with square and apothem.

Figure (ii) with cube and the apothem for three faces.

(d).

In terms of apothem the area of circle is .

Differentiate with respect to .

.

In terms of apothem the volume of the cube is .

Differentiate with respect to .

.

(e).

When the area of the sqyare is written in terms of apothem, the derivative is the formula for the perimeter of the square.

When the volume of the cube is written in terms of the apothems of its faces the derivative is the formula for the surface area of the cube.

(a).

and .

(b).

The derivative of the formula for the area of the circle is the formula for the circumference of the circle.

The derivative for the volume of the sphere is the formula for the surface area of the sphere.

(c).

(d).

and .

(e).

When the area of the sqyare is written in terms of apothem, the derivative is the formula for the perimeter of the square.

When the volume of the cube is written in terms of the apothems of its faces the derivative is the formula for the surface area of the cube.