Solving Derivatives with the Chain Rule

[arhzz]

Homework Statement

Denote c (t) the concentration of an active ingredient in the blood t hours after the injection
and c (t) = 16t applies
(10t + 20)^2. Find the maximum concentration and timing of the
occurrence.

Relevant Equations

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Hello! Now this is not really a physics problem of the usual kind but I'd say you could consider it one.Still I'd like to post my problem here because here I always get great help and advice.Now for this problem in particular,it is in the section of the book that deals with derivatives so I asummed I'd have to use it at some point.Now the derivative I used is c(t)/t.So t is our differantiation constant.Now here lies my problem. Since this is $\frac f g$ I used the rule. that is used which is $$\frac{f(x)' * g(x) - f(x) * g'(x)}{g(x)^2}$$ Now since I didnt get the same result,and after trying for about 40 minutes I went to look at the solution and they used the same rule (the numerator was the same) but they completely left the denomiantor, they simply derived without the g(x)^2 or in this case should be g(x)^4. Why is that so? What am I missing ?

Thanks

 

[BvU]

.. .and c (t) = 16t applies $\quad$ (10t + 20)^2 ...
What am I missing ?

I don't know what you are missing, but I miss a sensible problem statement

 

[BvU]

Now the derivative I used is c(t)/t.So t is our differantiation constant.Now here lies my problem.

From now on you want to type a space after a period
And type sensible sentences (the first two are not: c(t)/t is not a derivative and definitely not constant !)

Let me guess: you were given $$ c(t) = {16 t\over (10t+20)^2}\qquad ?$$ If I guessed right, what is your $f$ and what is your $g$ ?

 

[arhzz]

From now on you want to type a space after a period
And type sensible sentences (the first two are not: c(t)/t is not a derivative and definitely not constant !)

Let me guess: you were given $$ c(t) = {16 t\over (10t+20)^2}\qquad ?$$ If I guessed right, what is your $f$ and what is your $g$ ?

Yea I am given that, the thing is I am actually translating all of this from german,and it can be very tricky sometimes,especially because english is my 3rd language.I'll try to watch out on the phrases,and sentences sorry.

So as for your question, f is 16t and g should be $(10t + 20)^2$

 

[BvU]

Correct. So you see the $g^2 = (10t+20)^4$ coming ...

The exercise asks for a maximum. At a maximum, the derivative is zero. And a fraction is zero if the numerator is zero (and the denominator is non-zero, which needs to be checked, of course).
$10t+20$ is always positive and that is why they don't have to consider it.

 

[arhzz]

Yea I figured it would have to something with the way the exercise was asked,but thank you for the insight now I get it.But while I'm at it,what if 10t +20 wasnt always positive, if we had 10t-20 ? How would that change the exercise?

 

[BvU]

The patient would explode at t=2

 

[arhzz]

Oh wow, that I didnt expect.Thanks for your help!

 

[BvU]

You are welcome!