Differentiating a circle

  • Thread starter sonofjohn
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  • #1
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How would I go about finding the derivative of x2 + y2 = 36.

I know it is a circle with radius 6. Is there a better way to find the derivative then:

y2 = 36 - x2
y = (36 - x2)1/2
y = 1/2(36 - x2)-1/2(-2x)
y' = -x/(36-x2)1/2
 

Answers and Replies

  • #2
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Usually with cases like this where it is inconvenient to differentiate explicitly you can use implicit differentiation.
 
  • #3
James R
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Another way is implicit differentiation:

[tex]x^2 + y^2 = 36[/tex]

[tex]2x + 2y \frac{dy}{dx} = 0[/tex]

[tex]\frac{dy}{dx} = -\frac{x}{y} = -\frac{x}{\sqrt{36 - x^2}}[/tex]

Are you familiar with implicit differentiation?
 
  • #4
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You may use implicit differentiation. If f(x) = g(x), then f'(x) = g'(x). Therefore, we can have f(x) = x2 + y(x)2 (where I have written y explicitly as a function of x) and g(x) = 36.

Hence, we can differentiate both sides with respect to x without isolating y. Differentiating the left hand side, we get 2x + 2yy' using the chain rule. On the right hand side, differentiation 36, a constant function, just gives us 0.

We can then solve for y': 2x + 2yy' = 0 so y' = -x/y. Notice that we have y' in terms of both x and y(x) instead of just in x; this is a hallmark of implicit differentiation. If you solve for y in terms of x and plug it in, you find that

[tex]\frac{dy}{dx} = \frac{-x}{\pm\sqrt{36-x^2}}[/tex]​

depending on whether y was positive or negative. This is a more complete version of the answer you found by solving for y first.
 
  • #5
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Implicit differentiation. I have learned it, but obviously need to make more use of it. Thank you kind sirs.
 
  • #6
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On a second note, would it be better to leave the final answer in terms of x and y. Or should I solve for y and find y' in terms of x only. I prefer x/y but if the question asks for y' how should I answer?
 
  • #7
HallsofIvy
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1: You titled this "differentiation of a circle" which makes no sense. You cannot differentiate a geometric figure!

2: You then wrote "find the derivative of x2 + y2 = 36" which also makes no sense. You can differentiate (both sides of) an equation but you have to specify with respect to what variable.

3: Everyone here has assumed you really meant "find the derivative of y with respect to x, assuming that x2+ y2= 36".
 
  • #8
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1: You titled this "differentiation of a circle" which makes no sense. You cannot differentiate a geometric figure!

2: You then wrote "find the derivative of x2 + y2 = 36" which also makes no sense. You can differentiate (both sides of) an equation but you have to specify with respect to what variable.

3: Everyone here has assumed you really meant "find the derivative of y with respect to x, assuming that x2+ y2= 36".
Sorry for the mix up. Yes I meant solve for dy/dx.
 

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