The side of a cube increases at 1 cm / s. How fast is the diagonal of the cube changing when the side is 1 cm?
[itex] a^2+b^2=c^2 [/itex]
The Attempt at a Solution
I'm attempting to find the diagonal of the cube through Pythagorean Theory where I find the diagonal of one side of the cube to give me an equation for side length 'A', after finding side length 'A' I know that the height of the cube is 'x' centimeters so I can then again use Pythagorean Theory to find the equation of the diagonal going through the cube.
So I set all sides of the cube equal to 'x' and solve for a diagonal of one of the sides of the cube in order to get a side length ( 'A' ) of the diagonal within the cube. Using Pythagorean I come to the equation of:
[itex] \sqrt2x^2 = C [/itex]
After that I know that the height of the cube is 'x' so once again using Pythagorean I can derive an equation for the diagonal through the cube.
[itex](\sqrt2x^2)^2 + x^2 = D^2[/itex]
[itex]3x^2 = D^2[/itex]
[itex]\sqrt3x^2 = d[/itex]
Now this is where I become confused in order to find the rate of change at 1 cm. Do I then find the derivative of [itex]\sqrt3x^2 = D[/itex] through implicit differentiation?
[itex]D = \sqrt3x^2[/itex]
[itex]D' = (3x)/\sqrt3x^2 (dLength/dt)[/itex]
... I'm rather lost on what to do from here. Any help would be greatly appreciated.