# Find f"(x) given that 3x²+y²=2 using implicit differentiation?

Aug 17, 2015

$f ' ' \left(x\right) = - \frac{3 \left(3 {x}^{2} + {y}^{2}\right)}{y} ^ 3$

#### Explanation:

$3 {x}^{2} + {y}^{2} = 2$
Differentiate with respect to $x$:
$6 x + 2 y \frac{\mathrm{dy}}{\mathrm{dx}} = 0$
Differentiate once more with respect to $x$:
$6 + 2 y \frac{{d}^{2} y}{\mathrm{dx}} ^ 2 + 2 {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{2} = 0$

Re-arrange:
$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = - \frac{3 \left(3 {x}^{2} + {y}^{2}\right)}{y} ^ 3$