How do you use implicit differentiation to find (d^2y)/dx^2 of x^3+y^3=1

Donna Flynn 2022-07-23 Answered
How do you use implicit differentiation to find d 2 y d x 2 of x 3 + y 3 = 1
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Answers (1)

yermarvg
Answered 2022-07-24 Author has 19 answers
By implicitly differentiating twice, we can find
d 2 y d x 2 = - 2 x y 5
First, let us find d y d x
x 3 + y 3 = 1
by differentiating with respect to x
3 x 2 + 3 y 2 d y d x = 0
by subtracting 3 x 2
3 y 2 d y d x = - 3 x 2
by dividing by 3 y 2
d y d x = - x 2 y 2
Now, let us find d 2 y d x 2
by differentiating with respect to x
d 2 y d x 2 = - 2 x y 2 - x 2 2 y d y d x ( y 2 ) 2 = - 2 x ( y 2 - x y d y d x ) y 4
by plugging in d y d x = - x 2 y 2
d 2 y d x 2 = - 2 x [ y 2 - x y ( - x 2 y 2 ) ] y 4 = - 2 x ( y 2 + x 3 y ) y 4
by multiplying the numerator and the denominator by y
d 2 y d x 2 = - 2 x ( y 3 + x 3 ) y 5
by plugging in y 3 + x 3 = 1
d 2 y d x 2 = - 2 x y 5
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