Let's apply the implicit differentiation rule to the real world!

 

Here's a scenario in which the implicit differentiation rule and derivatives can be applied. 

 

Problem:

 

Ship A and Ship B are leaving a port at the same time, going in two different directions. After an hour, Ship A is going at 20 knots per hour, while Ship B is going at 15 knots per hour (Beyondships). The captain of Ship A wants to know how far Ship B is from them since they’re the only ship out there besides them. What will be the rate of change of distance between the two ships after one hour of sailing?

 

Solution: 

 

First, we must draw a diagram in order to find out what we know and don’t know about the problem.

 

 

 

 

 

 

 

 

 

 

 

 

 

We know that Ship A is going 20 knots per hour, and Ship B is going at 15 knots per hour. From the diagram, we can tell this is a right triangle, and the segment we want to solve is the hypotenuse.

 

 

 

 

 

 

From there, we know to use the Pythagorean Theorem in order to solve for "x", or the missing distance between the two ships. 

 

Then, we need to solve for x when t=1.

 

 

 

 

 

 

Now we know that x = 25 feet when t=1. However, in order to find the rate of change of the distance in between the two ships, we need to find the derivative equation of the Pythagorean Theorem. Tod do this, we have the apply implicit differentiation to the three variables as shown:

 

 

 

Since we want to find dx/dt, we have to divide each side by 2x.

 

 

 

 

Now all we have to do is plug the values in for the function in order to get dX/dt, or the change of the distance between the two ships.

 

 

 

 

 

 

 

 

 

This means that the distance between Ship A and Ship B is changing by 25 knots per hour.