Make-up #8

Related Rates FTW!!!

x and y are both differentiable functions of t and are related by the equation y=2x^3-x+4. When x=2 dx/dt=-1 Find dy/dt.

Some people are freaking out in their heads by now but not to fear, related rates problems are similar to optimization and implicit derivatives.

Steps for Related Rates Problems:

1. Identify all variables and equations.
2. Make a sketch and label. *VERY IMPORTANT*
3. Write an equation that involves your variables. *You can only have one unknown so a secondary equation may be given* (Usually this step is needed for word problems that ask you to find the rate at which a side of a triangle moves or the volume of a sphere changes.)
4. Take the derivative with respect to time, which is t.
5. Plug in all variables and solve.

Sure there are a lot of steps involved but they flow from one to the next so simply its like there is only 1 step: solve.

Step 1. Identify all variable and equations.

y=2x^3-x+4 x=2 dx/dt=-1 and dy/dt=?

Step 2. Make a sketch and label.

Um...since I am doing this on a blog I will not sketch and label the problem.

Step 3. Write an equation involving all variables.

Already given to us (y=2x^3-x+4)

Step 4. Differentiate with respect to "t".

dy/dt=6x^2(dx/dt)-1(dx/dt)

Step 5. Plug in a solve.

dy/dt=6(2^2)(-1)-1(-1)
dy/dt=-23

TA-DA!!! Is this not simple?