Use implicit differentiation to find $y^\prime$ and $y^{\prime\prime}$ if $y^{2}-x^{2}=5$. Express $y^{\prime\prime}$ in terms of $x$ and $y$.

Solution First, we assume there is a differentiable function $y=f( x)$ that satisfies $y^{2}-x^{2}=5$. Now we find $y^\prime$.



provided $y≠ 0$.

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Equations (3) and (4) both involve $y^\prime$. Either one can be used to find $y^{\prime\prime} .$ We use (3) because it avoids differentiating a quotient. $$\begin{eqnarray*} \dfrac{d}{dx} ( 2yy^\prime -2x) &=&\dfrac{d}{dx}0 \nonumber\\ \dfrac{d}{dx} ( yy^\prime ) -\dfrac{d}{dx}x &=&0 \nonumber\\ y\cdot \dfrac{d}{dx}y^\prime +\left( \dfrac{d}{dx}y\right) y^\prime -1 &=&0 \nonumber\\ yy^{\prime\prime} +( y^\prime ) ^{2}-1 &=&0 \nonumber\\ y^{\prime\prime} &=&\dfrac{1-( y^\prime ) ^{2}}{y}\tag{5} \end{eqnarray*}$$ provided $y≠ 0$. To express $y^{\prime\prime}$ in terms of $x$ and $y,$ we use (4) and substitute for $y^\prime$ in (5).