I did not understand this derivative -- help please

[cemtu]

Homework Statement

Mathematical Methods in Physics

Relevant Equations

Derivative

I have no idea how this derivative was taken.

 

[etotheipi]

It's wrong. $N_{0}$ is a constant (assuming the conventional interpretation of $N_0$ as the value of $N$ at $t=0$). $\frac{dN_0}{dt}$ is always going to be $0$ and doesn't really make any sense. And to go one further, you still need to derive $e^{-\lambda t}$ wrt $t$.

 

[cemtu]

dN0/dt = 2 is given thus N0 is 2t. There is no problem there sir.

 

[PeroK]

dN0/dt = 2 is given thus N0 is 2t. There is no problem there sir.

Your original post makes no sense. If you want any help you'll have to organise it into something meaningful.

 

[cemtu]

Okay right away.

 

[cemtu]

Your original post makes no sense. If you want any help you'll have to organise it into something meaningful.

I put the whole question and its solution.

 

[etotheipi]

Are you sure you don't mean $A_{0} = \left[-\frac{dN}{dt} \right]_{t=0} = 2$?

Of course, start off with the equation you wrote of $N = N_{0}e^{-\lambda t}$. Then you can derive both sides with respect to $t$ to obtain a relation pertaining to the activities.

How would you derive the RHS wrt $t$? Or perhaps more specifically, what can we do about the constant out the front?

 

[cemtu]

Are you sure you don't mean $A_{0} = \left[-\frac{dN}{dt} \right]_{t=0} = 2$?

Of course, start off with the equation you wrote of $N = N_{0}e^{-\lambda t}$. Then you can derive both sides with respect to $t$ to obtain a relation pertaining to the activities.

How would you derive the RHS wrt $t$? Or perhaps more specifically, what can we do about the constant out the front?

I know how to derive however our professor just replaced N with dN/dt (LHS) and N0 with dN0/dt (RHS). I don't know how. He didnt took the derivative as the way you and I want.

 

[etotheipi]

I know how to derive however our professor just replaced N with dN/dt (LHS) and N0 with dN0/dt (RHS). I don't know how. He didnt took the derivative as the way you and I want.

I can't speak for what your professor was trying to imply however I can say that $\frac{dN_{0}}{dt}$ is zero, as is the case for all constants. It's essentially a meaningless statement.

What do you get if you derive both sides of $N = N_0 e^{-\lambda t}$ with respect to $t$?

 

[cemtu]

I can't speak for what your professor was trying to imply however I can say that $\frac{dN_{0}}{dt}$ is zero, as is the case for all constants. It's essentially a meaningless statement.

What do you get if you derive both sides of $N = N_0 e^{-\lambda t}$ with respect to $t$?

Here

 

[PeroK]

Here

I suspect this might be an initial value problem. But, you've misunderstood what's been given as initial values.

Without seeing the problem stated clearly, we are all just guessing.

 

[etotheipi]

Here

You've done the differentiation right, $\frac{dN}{dt} = -\lambda N_{0} e^{-\lambda t} \implies A = A_{0}e^{-\lambda t}$ since $A = \lambda N$.

You must stop using $\frac{dN_{0}}{dt}$. It is wrong. I strongly suspect the intended meaning is the rate of change of $N$ at $t=0$, however we might just denote this (negative) $A_{0}$ (or that more clunky expression in post #7).

You should be able to just plug in the ratio of the activities.

 

[cemtu]

You've done the differentiation right, $\frac{dN}{dt} = -\lambda N_{0} e^{-\lambda t} \implies A = A_{0}e^{-\lambda t}$ since $A = \lambda N$.

You must stop using $\frac{dN_{0}}{dt}$. It is wrong. I strongly suspect the intended meaning is the rate of change of $N$ at $t=0$, however we might just denote this (negative) $A_{0}$ (or that more clunky expression in post #7).

You should be able to just plug in the ratio of the activities.

Sir, thank you.

 

[etotheipi]

Sir, thank you.

You're welcome, though no need to call me 'sir', I haven't met the Queen yet!

 

[Mark44]

Without seeing the problem stated clearly, we are all just guessing.

This...