Derivative of a log question

[Adi]

Hey guys I'm trying to do part c of this question and Im not sure if I'm doing it right can someone please tell me if what ive done so far is right?

 

[Adi]

This is what ive done

 

[JeffM]

It is very hard to follow this because you have two attachments to two different posts. But question c does not ask for the function but its derivative , no?

 

[Adi]

It is very hard to follow this because you have two attachments to two different posts. But question c does not ask for the function but its derivative , no?

Yes the question asks for its derivative. All I was trying to do in the 2nd attachment was to write out the function correctly before I could take it's derivative. I'm not sure if I've written it correctly. Also I find it confusing that the question asks for "the derivative" instead of several partial derivatives since there is more than one variable in the function. Any ideas?

 

[Adi]

a

Yes the question asks for its derivative. All I was trying to do in the 2nd attachment was to write out the function correctly before I could take it's derivative. I'm not sure if I've written it correctly. Also I find it confusing that the question asks for "the derivative" instead of several partial derivatives since there is more than one variable in the function. Any ideas?

am I supposed to take the derivative with respect to t?

 

[Dr.Peterson]

Yes. They explicitly ask for the derivative of the function ln(E(t)), which is a function of t, so you should find d/dt ln(E(t)). The only variable is t.

But your function is not quite right; you didn't raise all of p and r to the appropriate powers.

 

[JeffM]

a

am I supposed to take the derivative with respect to t?

Yes. You can take the derivative with respect to t because now the function now has only one independent variable, namely t. The variables p and r are gone having been replaced by functions of t: p_0 and r_0 are constants rather than variables. The notation is a bit confusing at first, but the equation has been parameterized.

As I said, it is hard for me to be sure whether or not you have done it correctly because I get dizzy flipping between different attachments.

 

[Adi]

Thanks guys. could one of you just write out what the function E(t) looks like for me? I'm so so so stuck

 

[HallsofIvy]

We are given that $E(p,r)= Ap^{-a}r^b$ and that $p(t)= p_0(1.06)^t$ and $r(t)= r_0(1.08)^t$. Putting those together, $E(t)=A[p_0(1.06)^t]^{-a}[r_0(1.08)^t]^b= Ap_0^{-a}r_0^b(1.06)^{-at}(1.08)^{bt}$.

But I suspect that the idea was to use the chain rule rather than writing it out like that.

 

[Adi]

We are given that $E(p,r)= Ap^{-a}r^b$ and that $p(t)= p_0(1.06)^t$ and $r(t)= r_0(1.08)^t$. Putting those together, $E(t)=A[p_0(1.06)^t]^{-a}[r_0(1.08)^t]^b= Ap_0^{-a}r_0^b(1.06)^{-at}(1.08)^{bt}$.

But I suspect that the idea was to use the chain rule rather than writing it out like that.

Thanks Hallsoflvy. How would you use the chain rule rather than writing it out like that?