anonymous
  • anonymous
Find the tangent line to the curve \[y=\cos(x+y)\] at the point \[(\Pi/2,0)\]
Calculus1
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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SolomonZelman
  • SolomonZelman
You need to find dy/dx of the curve
SolomonZelman
  • SolomonZelman
Do you have any trouble differentiating on both sides? (chain rule twice on the right side - once for the part inside the cosine - second chain for the y (the y') )
SolomonZelman
  • SolomonZelman
Ok, what is the derivative of y?

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anonymous
  • anonymous
Would it be y'?
SolomonZelman
  • SolomonZelman
yes exactly
anonymous
  • anonymous
So it's -sin(x+y) (y')
SolomonZelman
  • SolomonZelman
y is a function of x, (just as f(x) is) The derivative of f(x) is f'(x), AND the derivative of y is y'.
SolomonZelman
  • SolomonZelman
the derivative of x is 1, not 0
SolomonZelman
  • SolomonZelman
(Nice username btw, I like that)
anonymous
  • anonymous
Oh okay, I forgot the 1. \[-\sin(x+y)\times(1+y')=f'(x)\]
SolomonZelman
  • SolomonZelman
yes, y' = -sin(x+y) × (1+y')
anonymous
  • anonymous
Thanks!
SolomonZelman
  • SolomonZelman
Ok, now you need to isolate the y'
SolomonZelman
  • SolomonZelman
(y' is same as dy/dx)
SolomonZelman
  • SolomonZelman
\(\large\color{#000000 }{ \displaystyle y' = -\sin(x+y) \times (1+y') }\) \(\large\color{#000000 }{ \displaystyle y' = -\sin(x+y) -y'\sin(x+y) }\)
SolomonZelman
  • SolomonZelman
\(\large\color{#000000 }{ \displaystyle y' +y'\sin(x+y)= -\sin(x+y) }\)
SolomonZelman
  • SolomonZelman
\(\large\color{#000000 }{ \displaystyle y' (1+\sin(x+y))= -\sin(x+y) }\) \(\large\color{#000000 }{ \displaystyle y' =\frac{ -\sin(x+y) }{1+\sin(x+y)} }\)
SolomonZelman
  • SolomonZelman
I can't read that code clearly, what is the point of tangency? )
SolomonZelman
  • SolomonZelman
\((\pi/2~,~0)\) that?
anonymous
  • anonymous
yeah
SolomonZelman
  • SolomonZelman
So, plug that point \(x=\pi/2\) and \(y=0\) into the derivative, and see what you get.
anonymous
  • anonymous
x=-1/2
SolomonZelman
  • SolomonZelman
x ?
SolomonZelman
  • SolomonZelman
you mean the result is -1/2?
anonymous
  • anonymous
Oh I meant y'
SolomonZelman
  • SolomonZelman
yeah :)
SolomonZelman
  • SolomonZelman
-1/(1+1)=-1/2 (because sin(π/2)=1)
SolomonZelman
  • SolomonZelman
So your slope is -1/2. And the point is \((\pi/2,0)\) you got it from here
anonymous
  • anonymous
Ok, thank you very much!!
SolomonZelman
  • SolomonZelman
If you would like to later confirm the answer, then, here is the graph of the curve, the point, and the tangent line. https://www.desmos.com/calculator/hdtir0corc
anonymous
  • anonymous
Alright, thanks! :)
SolomonZelman
  • SolomonZelman
Good Luck!

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