`y^2 = (x^2 - 49)/(x^2 + 49), (7,0)` Find `dy/dx` by implicit differentiation and evaluate the derivative at the given point.

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Chapter 2, 2.5 - Problem 23 - Calculus of a Single Variable (10th Edition, Ron Larson).
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gsarora17 | (Level 2) Associate Educator

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`y^2=(x^2-49)/(x^2+49)`

Differentiating both sides with respect to x,

`2ydy/dx=((x^2+49)d/dx(x^2-49)-(x^2-49)d/dx(x^2+49))/(x^2+49)^2`

`2ydy/dx=((x^2+49)(2x)-(x^2-49)(2x))/(x^2+49)^2`

`2ydy/dx=(2x(x^2+49-x^2+49))/(x^2+49)^2`

`2ydy/dx=(2x(98))/(x^2+49)^2`

`dy/dx=(98x)/(y(x^2+49)^2)`

Now derivative at the point (7,0) can be evaluated by plugging in the value of x,y in dy/dx.

Since y=0 and it is in the denominator of dy/dx , so the derivative at the point(7,0) does not exist.

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