Chain Rule: Price P(c) = 2c - 18/c, where c(t) = 9 + 3/t is production cost. Find dP/dt at t = 3.

[miggs5434]

Could someone kindly help me with this question?

The price P of an electronic component is known to be: P(c) = 2c - 18/c where c is the manufacturer's cost to produce it. The cost c is a function time t according to
c(t) = 9 + 3/t Determine the rate of change of price with respect to time (dP/dt) at time t = 3 years.

 

[Dr.Peterson]

Welcome to FMH.

As you will see when you read our posting guidelines, we want you to show us your work, so we can tell what help you need. Here, you clearly know that you need to use the chain rule; have you done that? What did you get for dP/dc and dc/dt? What difficulty did you find?

 

[HallsofIvy]

If P(c)= 2c- 18/c then [math]P(t)= 2t- \frac{18}{t}= 2t- 18t^{-1}[/math]. Can you differentiate that?

 

[miggs5434]

I have tried to solve this problem but I keep going in circles...I have no trouble differentiating, I just don't know where to start. I'm a bit uncertain as to the relationship between the equations; I'm not sure how to combine them.
Here is what I have so far that I know is at least correct, the derivations of the equations:
P(c) = 2 + 18/c^2
c(t) = -3/t^2
P(t) = 2 +18/t^2

 

[Dr.Peterson]

I have tried to solve this problem but I keep going in circles...I have no trouble differentiating, I just don't know where to start. I'm a bit uncertain as to the relationship between the equations; I'm not sure how to combine them.
Here is what I have so far that I know is at least correct, the derivations of the equations:
P(c) = 2 + 18/c^2
c(t) = -3/t^2
P(t) = 2 +18/t^2

You don't want to do anything with P(t); the problem is about the composite function P(c(t)). By the chain rule, you know that dP/dt = dP/dc * dc/dt, that is P'(c(t))*c'(t).

As I asked before, what do you get for dP/dc and dc/dt? I guess where you wrote above P(c) and c(t), you meant P'(c) and c'(t); am I right? If so, you're correct. Now find their values for t = 3, and multiply.

 

[HallsofIvy]

You have written "P(c)= 2+ 18/c^2" and "c(t)= -3/t^2" but in your initial post you wrote "P(c)= 2c- 18/c" and "c(t)= 9+ 3/t". You mean "P'(c)= 2+ 18/c^2" and "c'(t)= -3/t^2", the derivatives of P and c rather than P and c themselves. I hope you understand the difference but carelessness like that can cause serious errors! Using the chain rule, the derivative of P with respect to t is (dP/dc)(dc/dt)= (2+ 18/c^2)(-3/t^2). At t= 3, c(3)= 9+ 3/3= 9+ 1= 10. So (2+ 18/10^2)(-3/3^2)= (2.18)(-1/3)= -0.72666....