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Alina [70]
3 years ago
9

Find y'' by implicit differentiation. 4x3 + 5y3 = 2

Mathematics
1 answer:
Veronika [31]3 years ago
8 0
Differentiate implicitly using the power rule: 

<span>12x^2 + 15y^2y' = 0 </span>

<span>Make y' the subject: </span>

<span>15y^2y' = -12x^2 </span>
<span>y' = -(4x^2) / 3y^2 </span>

<span>Differentiate again using the quotient rule: </span>

<span>y'' = [-8x(3y^2) - 6yy'(-4x^2)] / 9y^4 </span>
<span>y'' = (-24xy^2 + 24x^2yy') / 9y^4 </span>

<span>Simplify: </span>

<span>y'' = (-8xy + 8x^2y') / 3y^3 </span>

<span>Substitute (4x^2) / 3y^2 back for y': </span>

<span>y'' = (-8xy + 8x^2 * 4x^2 / 3y^2) / 3y^3 </span>
<span>y'' = (-24xy^3 + 32x^4) / 9y^5</span>
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