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: a^{2} = c^{2} - b^{2}
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.
a^{2} = c^{2} - b^{2}
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.)