Graphical Understanding of Derivatives:

A ladder 26 feet long is leaning against a wall. The ladder begins to move such that the bottom end moves away from the wall at a constant velocity of 2 feet by per second.   What is the downward velocity of the top end of the ladder when the bottom end is 10 feet from the wall?

Solution:

Begin with the Pythagorean Theorem for a right triangle:     a² = c² - b²

Obtain the derivative of both sides of this equation with respect to time t.  The c, representing the length of the ladder is a constant.

2a(da/dt) = -2b(db/dt)

a(da/dt) = -b (db/dt)

But, db/dt is the velocity at that the bottom end of the ladder is moving  away  from  the  wall,  equal  to  2  ft/s,  and  da/dt  is  the downward  velocity  of the top end of the ladder  along  the wall, that is the quantity  to be determined.  Set b equal to 10 feet, substitute the known values into the equation, and solve for a.

a² = c² - b²

a= 24 ft

a(da/dt) = -b (db/dt)

(da/dt) = -b/a (db/dt)

(da/dt) = -10 ft/24 ft(2 ft/s)

(da/dt) = -0.833 ft/s

Therefore, when the bottom of the ladder is 10 feet from the wall and moving at 2ft/sec., the top of the ladder is moving downward at 0.833 ft/s. (The negative sign denotes the downward direction.)