Time derivative Question

4football

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Hi,

I have been struggling with this question for a while now and I don't know if im on the right track or where to go from here.

Question

Show that the time derivative of the logarithm of a variable equals its growth rate measured in percent. Use the following notation for a time derivative:
dX/dT =X

My workings

dx/dt = time variable
ln=logarithm
X=Vairable
@x/x=growth rate in percentage terms

let change in variable = @ because I can't find the triangle symbol.

dx/dt(ln)X=@X/X


I feel im getting quite lost or on the wrong track for this question. Any help would be greatly appreciated!!

Kind regards,

James
 
Question: Show that the time derivative of the logarithm of a variable equals its growth rate measured in percent. Use the following notation for a time derivative:
dX/dT =X

My workings:

dx/dt = time variable
ln=logarithm
X=Vairable
@x/x=growth rate in percentage terms

let change in variable = @ because I can't find the triangle symbol.

dx/dt(ln)X=@X/X
You are given a variable, X, and are asking to show that two things are equal: the derivative, with respect to time t, of the log of X; and the growth rate of X, written in percentage form. So work in steps on each side of this proposed equality.

a) What is the variable?
b) What is the log of that variable?
c) What is the derivative, with respect to time, of the log of that variable?

d) What is the (derivative-based) expression for the growth rate of the variable?
e) What is this growth rate, expressed as a percent (that is, involving division in some manner)?

Please reply with your responses. Thank you! ;)
 
a) What is the variable?

The variable is X

b) What is the log of that variable?

the log of the variable is ln(x)

c) What is the derivative, with respect to time, of the log of that variable?

the derivative of d/dx[ln(x)] = 1/x

d) What is the (derivative-based) expression for the growth rate of the variable?

I get more stuck here, and need this to solve e)

e) What is this growth rate, expressed as a percent (that is, involving division in some manner)?

[ln(x+@x)/@x) ?

I don't know how to work out d) and e)

Again any help would be appreciated!! :)

Thanks in advance,

James
 
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a) What is the variable?

The variable is X
Okay; we'll stick with the given upper-case "X", rather than lower-case "x".

b) What is the log of that variable?

the log of the variable is ln(x)
Almost: ln(X)

c) What is the derivative, with respect to time, of the log of that variable?

the derivative of d/dx[ln(x)] = 1/x
No; this is the derivative with respect to "x", which isn't even the variable we're using.

Instead, let's stick to what they gave you (and what you specified above):

ln(X), where X is (apparently, according to this exercise) a function of time t, so it's really ln(X(t)).

Then the derivative, using the Chain Rule, would be... what?

d) What is the (derivative-based) expression for the growth rate of the variable?

I get more stuck here, and need this to solve e)
If the derivative gives you the rate of change (with respect to the variable you're differentiating in terms of), then what is the expression for the growth rate (that is, the [positive] rate of change of) the variable X = X(t)? ;-)
 
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