help-derivatives-ex

Question

Answer 1

(1)

Step 1 :

Thin sheet of ice is in the form of a circle.

Area of the circle is , where is the radius of the circle.

Area of the sheet is decreasing at rate of m²/sec.

Consider .

Differentiate on each side with respect to .

Step 2:

Determine the radius when the area of the sheet is m².

Determine the rate of change in radius when the area of the sheet is m².

Substitute and in .

Therefore, the radius of the sheet is decreasing at 0.0498 m/sec. 

Solution :

The radius of the sheet is decreasing at 0.0498 m/sec.

Answer 2

(2)

Step 1:

The function is .

Apply derivative on each side with respect to .

Power rule of derivatives : .

.

.

Step 2:

.

Apply derivative on each side with respect to .

 

.

Step 3:

.

Apply derivative on each side with respect to .

. 

Solution :

The third derivative of the  function is .

Answer 3

(3)

Step 1:

The function is .

Apply derivative on each side with respect to .

Product rule of derivatives : .

Power rule of derivatives : .

Derivative of the cosine function:.

.

Step 2:

.

Apply derivative on each side with respect to .

 

.

Solution :

The second derivative of the  function is .