Consider the equation 8x^2 − y^2 = 9 (a) Find y' by implicit differentiation.(b) Solve the equation explicitly for y and differentiate to get y' in terms of x.

Answer 1

8x^2 - y^2 = 9

8x^2 - 9 = y^2

sqrt(8x^2 - 9) = y

y = (8x^2 - 9)^(1/2)

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implicit differentiation:

16x - 2yy' = 0

8x - yy' = 0

8x = yy'

8x/y = y'

8x/(sqrt(8x^2 - 9)) = y'

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Differentiating:

y' = (1/2)(8x^2-9)^(-1/2) ( 16x)

= 8x / sqrt(8x^2-9)