#help pls

185 messages · Page 1 of 1 (latest)

daring tiger
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On 1 January 2000, 80 mg of a radioactive substance was placed in a monitoring station. The amount y mg decreases with time t years. The half-life is 25 years.

a) Determine y(t) and draw the graph for the years 2000-2100.
b) Determine and interpret y′(50).
Answer to one decimal place.

nocturne drumBOT
daring tiger
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help please

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<@&286206848099549185>

balmy tartan
daring tiger
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y= 80*0.9726^x

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now i am on b)

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i am trying to find y'(50)

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helppp

balmy tartan
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do you know how to find the derivative of a^x when "a" is a constant

daring tiger
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e

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idk

balmy tartan
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you can use logarithmic differentiation

daring tiger
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pls can u help me quick

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e^x

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?

balmy tartan
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set y = a^x

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take the ln of both sides

daring tiger
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pls

balmy tartan
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yes, and the problem you have is you dont know how to differentiate .9276^x

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are you in a hurry

daring tiger
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Yeah kinda

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i am stuck on this problem for 2 hours

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so i am tired

balmy tartan
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ok let me just show you how to find the derivative of a^x first

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so you can rederive it later, thats what i do

daring tiger
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okay sure

balmy tartan
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if y = a^x, you want dy/dx

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so if you take the ln of both sides, you get lny = ln(a^x) = xlna from your logarithm rules

daring tiger
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oh

balmy tartan
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so what happens if you take the derivative with respect to x now

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of lny = xlna

daring tiger
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lny goes away?

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x goes away

balmy tartan
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well whats the derivative of lny with respect to x

daring tiger
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lna stays

daring tiger
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x goes away

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lny = lna

balmy tartan
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well you take the derivative of both sides

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so (the derivative of lny) = lna

daring tiger
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wait

balmy tartan
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y is a function of x, so y = y(x)

daring tiger
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ln = lna

balmy tartan
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well ln isnt a number so it needs to have something inside

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whats the derivative of lnu

daring tiger
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lna u mena

balmy tartan
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if u = u(x)

daring tiger
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oh

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u'(x)

balmy tartan
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whats the derivative of ln(u)

daring tiger
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u goes away

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ln

balmy tartan
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no, the ln isnt a number, its like a square root, it needs something inside, so only ln doesnt make sense just like a square root with no number doesnt make sense

daring tiger
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ok

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can u just show me please?

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i already have the answer in my book

balmy tartan
daring tiger
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damn

balmy tartan
daring tiger
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i have never seen this

daring tiger
balmy tartan
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times du/dx, because of the chain rule

daring tiger
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i have not learned this

balmy tartan
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because u is a function of x

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thats your teachers fault then

daring tiger
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yeah

balmy tartan
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well anyways the derivative of lnu is (1/u) (du/dx)

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so back to the original question

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the derivative of both sides of lny = xlna

daring tiger
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1/y = 1/a

balmy tartan
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y is a function of x so you would need to multiply the left side by ...

daring tiger
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lna

balmy tartan
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dy/dx

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do you know the chain rule

daring tiger
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no

balmy tartan
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are you sure

daring tiger
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can i show u how my book solved it?

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bc ur teaching me a different method

balmy tartan
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i dont know how else you would do it but sure

daring tiger
balmy tartan
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your book rewrote the the 2^(-t/25) as e^(-ln2 * t/25)

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you get the same derivative either way

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but going off your book, whats the derivative of e^(-ln2 * t/25)

daring tiger
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which method is easiest yours or the book?

balmy tartan
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i think mine, bc you can just find the derivative of 2^x straight up with no rewriting, but if you havent learned it yet probably your book will work for now

daring tiger
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oh

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i dont get the book method either

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so

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🙁

balmy tartan
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whats the derivative of e^x

daring tiger
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lnex

balmy tartan
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no its just e^x

daring tiger
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y = e^x

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then

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x = lny

balmy tartan
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1 = (dy/dx) (1/y)

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so dy/dx = y = e^x

daring tiger
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i am so confuseddd

balmy tartan
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lets start with the chain rule

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if you have a function within a function

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the derivative of the whole thing is the derivative of the outer times the derivative of the inner

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so the derivative of (2x+5)^5 with respect to x

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is 5(2x+5)^4 (2)

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5(2x+5)^4 is the derivative of the outer

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2 is the derivative of the inner

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alright so the derivative of e^x is e^x

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but the derivative with respect to x of e^(u(x)) is (e^(u(x)))(du/dx)

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because u is a function of x

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so, for the function e^(-ln2 * t/25)

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what is u(t) in this case

daring tiger
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idk

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🙁

balmy tartan
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if you have e^(2x)

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u is 2x

daring tiger
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yeah so

balmy tartan
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because you know the derivative of e^u

daring tiger
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2e^2x

balmy tartan
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exactly

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so for e^(-ln2 * t/25), what is e to the power to

daring tiger
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but here i have y= 80 * 0.9726^t

balmy tartan
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its all the same thing, just different ways of writing it

daring tiger
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so i understand

balmy tartan
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but you have to take the derivative however you write it, so would you rather take the derivative of a^x or e^u?

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i was gonna do a^x but your book is doing is e^u

daring tiger
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e^u

balmy tartan
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ok so that would be the book method

daring tiger
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wait

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is a^x easier?

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like more quick

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then i ll take it

balmy tartan
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do you want me to show you how to get the derivative of a^x without memorising a formula

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you can apply the method to more general functions if you need to later on

daring tiger
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okayy

balmy tartan
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ok

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so if "a" is a constant and y = a^x , you want dy/dx

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so rewriting, you get ln(y(x)) = xlna

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the derivative of the right is lna , and the derivative of the left is (dy/dx)(1/y), the dy/dx comes from the chain rule

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so (dy/dx)(1/y) = lna

daring tiger
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acc i am confusedd sorry

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this is just new for me

balmy tartan
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what are you confused about

balmy tartan
daring tiger
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mhmhm

balmy tartan
daring tiger
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ok i want to learn my book method instead

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sorry

balmy tartan
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ok but lets not switch again, itll make it more confusin

daring tiger
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okay

balmy tartan
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so the book method

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e^(-ln2 * t/25) is the function

daring tiger
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why

balmy tartan
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what do you think u(t) should be, if you know the derivative e^u(t) with respect to t is du/dt * e^u(t)

balmy tartan
daring tiger
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i dont get this part

image.png
balmy tartan
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whats e^ln2

daring tiger
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like why write t instead of 1

daring tiger
daring tiger
balmy tartan
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what does ln2 mean in words

balmy tartan
daring tiger
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ok

balmy tartan
daring tiger
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idk

daring tiger
balmy tartan
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in order to understand that you have to understand the math ladder your book uses, and so you have to start at the bottom of the ladder

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first step is understanding ln2

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if ln2 = a

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e^a = 2

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basically, its the number for which e to the power of the number equals the number in the log

daring tiger
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mhm

balmy tartan
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so log base 3 of 9 is 2

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because 3^2 = 9

daring tiger
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ok

balmy tartan
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so what is e^ln2

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first step is understanding ln2
if ln2 = a
e^a = 2

daring tiger
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2

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ln2 i mean

balmy tartan
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no you were right the first time

daring tiger
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oh