# Homework Help: Calculuate rate of change

1. Dec 2, 2007

### PlasmaSphere

1. The problem statement, all variables and given/known data

The temparature T of a liquid at a time t minutes is given by the equation
T = 30 + 20e-0.05t, for t > 0

(1)Write down the initial tempatature o the liquid, and find the intial rate of change of temparature.

(2)Find the time at which the temparature is 40

2. Relevant equations

3. The attempt at a solution

When t=0 the initial temparture is 50. I know that to have to find dT/dt, and i and looked at the answer, which says dT/dt = -0.05 x 20e-0.05t, but i cant see how they got that. what happened to the 30? and how did they get to that point?

2. Dec 2, 2007

### kevinr

They took the derivative of your T equation. Thus derivative of 30 = 0.

3. Dec 2, 2007

### PlasmaSphere

ok, cheers.

I still dont see how we get from T = 30 + 20e-0.05t to dT/dt = -0.05 x 20e-0.05t. i thought that the power had to decrease by 1 when you differenciated it and then multiply by the old power, so 20e-0.05 should become -0.05 x 20e-1.05. but thats not what the answer says, which is why i'm confused.

Last edited: Dec 2, 2007
4. Dec 2, 2007

### hotcommodity

If I have a function $$f(x) = e^u$$, where u can be any kind of x term, say .05x for instance, then I know that $$d/dx = e^u u'$$ where u' is the derivative of u, or in this case, the derivative of .05x, which is just .05. Does that make sense?

5. Dec 2, 2007

### kevinr

Well there are two because thats how to find derivative of e.

If you had e^2x, than the derivative would be 2e^2x. (2 comes from derivative of 2x).

(if you have e^x, than it is 1e^x = e^x because derivative of x is 1)

So in your problem, what is the derivative of -0.05t? It is -0.05 and thus that is why it comes in the front.

If you are confused ill try to explain better. Let me know!

(Sorry i dont know special writing method yet)

Last edited: Dec 2, 2007
6. Dec 2, 2007

You are trying to apply the power rule, but you have $$e^x$$ not $$x^n$$ the base is e and the exponent is a function,

...chain rule. $$\frac{d}{dx}[e^u]=e^u*\frac{du}{dx}$$ if I remember correctly.

Casey

Edit: I see that like nine others were typing at the same time as me.when it rains it pours! Take your pick :)

7. Dec 2, 2007

### PlasmaSphere

Sorry, i eddited my question after you answered it. However i think i get it now, I just forgot that e is its own deriviative.

so when i said; "i thought that the power had to decrease by 1 when you differenciated it and then multiply by the old power, so 20e-0.05 should become -0.05 x 20e-1.05.", thats not correct because you differenciate e differently

so it should become -0.05 x 20e-0.05t

8. Dec 2, 2007

### PlasmaSphere

thanx, thats what i needed. I understand now.