Thread: Rate Problem

Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. If V is the volume of such a cube with side length x, calculate the derivative when x = 5 mm.

V '(5) = ?


What does this quantity represent?
V '(5) represents the rate at which the volume is increasing with respect to the side length as V reaches 15 mm3.
V '(5) represents the rate at which the volume is increasing as x reaches 15 mm.
V '(5) represents the rate at which the volume is increasing with respect to the side length as x reaches 5 mm.
V '(5) represents the rate at which the side length is increasing with respect to the volume as x reaches 5 mm.
V '(5) represents the volume as the side length reaches 5 mm.

(Attempt)

I dont' know wat to do? Do I need to do the derivatives? Please show me! Thank you!

First, we use the formula for the volume of a cube:



We are asked to find the derivative of with respect to and evaluate it at . To do this, we may either use the Power Rule:



or we may use the definition of derivative:



The derivative measures the rate of change of the function with respect to its independent variable. If is considered a function of , then the derivative of basically measures the relative change of when we change by a very small amount.