Tutorial Exercise The length of a rectangle is increasing at a rate of 6 cm/s and its width is increasing at a rate of 7 cm/s. When the length is 10 cm and the width is 7 cm, how fast is the area of the rectangle increasing (in cmz/s)? Step 1 Let / and w represent the length and width of the rectangle in cm, and let A represent the area of the rectangle in cm . Writing an equation for A in terms of / and w gives the following result. A = W . I lw Step 2 We need to determine how fast the area of the rectangle is increasing. In other words, we are looking for a rate of change of the area. In this problem, the volume, the length, and the width are all functions of the time t, where t is measured in seconds. The rate of increase of the area with respect to time is the derivative . The rate of increase of the length and width with respect to time are the derivatives - - and aw dt dt , respectively. Therefore, we need to differentiate each side of our area equation with respect to t using the product rule. Doing so gives the following result where is measured in cm2/s. dA d(lw) dt dt dA dw + dt dt dt