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Calculus/quick question about implicit differentiation

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Expert: - 10/16/2009


Question
Implicit differentiation. Calculate y' implicitly

a.9x^2 +4y^2 -36=0

b.Find the slope of the tangent when x= 4/3

Answer
a. To find y', we can differentiate the equation as is.
It is 18x + 8yy' = 0, so y' = -8x/(8y) = -9x/(4y).

b. When x is 4/3, we have 9(4/3)² + 4y² - 36 = 0.
Note that 9(4/3)² = 9*16/9 = 16.
Since 16-36 is -20, add 20 to both sides.
This gives 4y² = 20, so y = √5.

Since taking the derivative gave 18x + 8yy' = 0, we can put in x and y.
18(4/3) + 8√5•y' = 0, or 24 + 8√5•y' = 0.  Divide by 8, giving 3 + √5•y' = 0.
This says the √5•y' = -3, so y' = -3√5/5.

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