Differentiation with chain rule correct?

[squawkasian]

Can't seem to figure out if I am doing the differentiation right for this word problem. Here's the problem.

In the mysterious lost city of Mim, the length of daylight (in hours) on the tth day of the year is modeled by the function

L(t) = 12 + sin[ (2pi/365)*(t-80)]

Use this model to compare how the number of hours of daylight is increasing on March 20 and June 13 (assume that this is a standard year, not a leap year).

Rate of increase on March 20 = ?
Rate of increase on June 13 = ?

So basically what I've done is this:

L'(t) = [12]' + { sin[ (2pi/365)*(t-80) ] }'
= 0 + cos[ (2pi/365)*(t-80) ] * (t-80)'
= 0 + cos[ (2pi/365)*(t-80) ] * (1 - 0)'
= cos[ (2pi/365)*(t-80) ]

So after that I plug in 79 ( March 20 is the 79th day of the year) and get 0.999851839209116.
As well as 164 ( June 13 is the 164th day of the year ) and get 0.124479263886789.
I'm doing homework online that checks if the answer is correct and my answers are incorrect.

Is my differentiation correct or am I doing something not quite right? Or maybe my approach to the word problem is incorrect? Any help would be appreciated.

 

[Deleted member 4993]

Can't seem to figure out if I am doing the differentiation right for this word problem. Here's the problem.

In the mysterious lost city of Mim, the length of daylight (in hours) on the tth day of the year is modeled by the function

L(t) = 12 + sin[ (2pi/365)*(t-80)]

Use this model to compare how the number of hours of daylight is increasing on March 20 and June 13 (assume that this is a standard year, not a leap year).

Rate of increase on March 20 = ?
Rate of increase on June 13 = ?

So basically what I've done is this:

L'(t) = [12]' + { sin[ (2pi/365)*(t-80) ] }'
= 0 + 2pi/365 * cos[ (2pi/365)*(t-80) ] * (t-80)'
= 0 + cos[ (2pi/365)*(t-80) ] * (1 - 0)'
= cos[ (2pi/365)*(t-80) ]

So after that I plug in 79 ( March 20 is the 79th day of the year) and get 0.999851839209116.
As well as 164 ( June 13 is the 164th day of the year ) and get 0.124479263886789.
I'm doing homework online that checks if the answer is correct and my answers are incorrect.

Is my differentiation correct or am I doing something not quite right? Or maybe my approach to the word problem is incorrect? Any help would be appreciated.

 

[squawkasian]

Thanks a lot for the help. Could you tell me the reason for adding the 2pi/365 and multiplying it by the stuff I've done?

 

[mmm4444bot]

Thanks a lot for the help. Could you tell me the reason for adding the 2pi/365 and multiplying it by the stuff I've done?

Hello Asian Squawk:

Firstly, be careful how you employ the verb "to add".

To answer your question, the Chain Rule tells us that after we differentiate the sine function, we must multiply the result by the derivative of the sine function's argument, because this argument itself is also a function in t.

If you calculate the derivative of the sine function's argument, you will get 2*Pi/365.

Perhaps, you might have realized this if you had written out your step as follows.

0 + cos[ (2pi/365)*(t-80) ] * [2*Pi/365*(t-80)]'

If you're still not sure, then carry out the multiplication by (t-80) above BEFORE you calculate the indicated derivative.

Cheers,

~ Mark