Answer
$y''=-\dfrac{5x^{4}}{y^{11}}$
Work Step by Step
Find $y''$ if $x^{6}+y^{6}=1$
Use implicit differentiation to evaluate the first derivative:
$x^{6}+y^{6}=1$
$6x^{5}+6y^{5}y'=0$
Solve for $y'$:
$y'=-\dfrac{6x^{5}}{6y^{5}}$
$y'=-\dfrac{x^{5}}{y^{5}}$
Apply implicit differentiation again (and the quotient rule) to obtain the second derivative:
$y''=-\dfrac{(y^{5})(x^{5})'-(x^{5})(y^{5})'}{y^{10}}=-\dfrac{5x^{4}y^{5}-5x^{5}y^{4}y'}{y^{10}}=...$
Substitute $y'$ by $-\dfrac{x^{5}}{y^{5}}$ and simplify:
$...=-\dfrac{5x^{4}y^{5}-5x^{5}y^{4}\Big(-\dfrac{x^{5}}{y^{5}}\Big)}{y^{10}}=...$
$...=-\dfrac{5x^{4}y^{5}+\dfrac{5x^{10}}{y}}{y^{10}}=-\dfrac{\dfrac{5x^{4}y^{6}+5x^{10}}{y}}{y^{10}}=-\dfrac{5x^{4}y^{6}+5x^{10}}{y^{11}}=...$
$...=-\dfrac{5x^{4}(x^{6}+y^{6})}{y^{11}}=\displaystyle-\frac{5x^4}{y^{11}}$
