$\mathbf{In}\mathbf{}\mathbf{Problem}\mathbf{}\mathbf{7}\mathbf{}\mathbf{find}\mathbf{}{\mathbf{d}}^{\mathbf{2}}\mathbf{y}\mathbf{/}{\mathbf{dx}}^{\mathbf{2}}\mathbf{}\mathbf{at}\mathbf{}\left(3,-1\right)\mathbf{.}$

$\mathrm{They}\mathrm{provide}\mathrm{an}\mathrm{equation}\mathrm{or}\mathrm{curve}\mathrm{that}\mathrm{is}{\mathrm{y}}^3-{\mathrm{x}}^2\mathrm{y}=8.$

To find the slopes of tangents to curves that aren't clearly functions, we can utilize implicit differentiation (they fail the vertical line test). Parts of yare assumed to be functions that satisfy the supplied equation but yis not a function ofx.

Consider the equation of the curve below:

$y^3-x^{2}y=8$

Differentiate the provided above equation of curve with respect to x,

$$\begin{gathered} 3y^2\frac{dy}{dx}-\left(x^2\frac{dy}{dx}+y.2x\right)=0 \\ 3y^2\frac{dy}{dx}-x^2.\frac{dy}{dx}-2xy=0 \\ \left(3y^2-x^2\right)\frac{dy}{dx}-2xy=0 \end{gathered}$$

Hence,

$\frac{dy}{dx}=\frac{2xy}{3y^2-x^2}$

Again, differentiate the provided above equation of a curve,

$$\begin{gathered} \frac{d^{2}y}{d{x}^2}=\frac{\left(3y^2-x^2\right)\left[2\left(x{\displaystyle \frac{dy}{dx}}+y\right)\right]-2xy\left[3\left(2y\right){\displaystyle \frac{dy}{dx}}-2x\right]}{{\left(3y^2-x^2\right)}^2} \\ =\frac{\left(3y^2-x^2\right)\left[2x\frac{dy}{dx}+2y\right]-2xy\left[6y\frac{dy}{dx}-2x\right]}{{\left(3y^2-x^2\right)}^2} \\ =\frac{\left(3y^2-x^2\right)\left[2x\frac{dy}{dx}+2y\right]-\left[12x{y}^2\frac{dy}{dx}-4x^{2}y\right]}{{\left(3y^2-x^2\right)}^2} \end{gathered}$$

Then,

role="math" localid="1658821013915" $$\begin{gathered} \frac{d^{2}y}{d{x}^2}=\frac{4x^{2}y+\left(3y^2-x^2\right)\left(2y\right)-\left[{\displaystyle \frac{12x^2{y}^3}{3y^2-x^2}}-4x^{2}y\right]}{{\left(3y^2-x^2\right)}^2} \\ =\frac{6x^2+6y^3-2x^{2}y-{\displaystyle \frac{24x^2{y}^3}{3y^2-x^2}}+4x^{2}y}{{\left(3y^2-x^2\right)}^2} \end{gathered}$$

Put the (3,-1) For (x,y) ,

$$\begin{gathered} {\left(\frac{d^{2}y}{d{x}^2}\right)}_{\left(3,-1\right)}=\frac{6{\left(3\right)}^2\left(-1\right)+6y^3-2{\left(3\right)}^2\left(-1\right)-{\displaystyle \frac{24{\left(3\right)}^2{\left(-1\right)}^3}{3{\left(-1\right)}^2-{\left(3\right)}^2}}+4{\left(3\right)}^2\left(-1\right)}{{\left(3{\left(-1\right)}^2-{\left(3\right)}^2\right)}^2} \\ =\frac{-96}{36} \\ =\frac{-8}{3} \end{gathered}$$

$$\begin{gathered} y^3-x^{2}y=8 \\ {\left(\frac{d^{2}y}{d{x}^2}\right)}_{\left(3,-1\right)}=\frac{-8}{3} \end{gathered}$$

$$\begin{gathered} 3y^2\frac{dy}{dx}-\left(x^2\frac{dy}{dx}+y.2x\right)=0 \\ 3y^2\frac{dy}{dx}-x^2.\frac{dy}{dx}-2xy=0 \\ \left(3y^2-x^2\right)\frac{dy}{dx}-2xy=0 \\ \frac{dy}{dx}=\frac{2xy}{3y^2-x^2} \\ \frac{d^{2}y}{d{x}^2}=\frac{\left(3y^2-x^2\right)\left[2\left(x\frac{dy}{dx}+y\right)\right]-2xy\left[3\left(2y\right)\frac{dy}{dx}-2x\right]}{{\left(3y^2-x^2\right)}^2} \\ =\frac{\left(3y^2-x^2\right)\left[2x\frac{dy}{dx}+2y\right]-2xy\left[6y\frac{dy}{dx}-2x\right]}{{\left(3y^2-x^2\right)}^2} \\ =\frac{3y^2-x^2\left[2x\frac{dy}{dx}+2y\right]-\left[12x{y}^2\frac{dy}{dx}-4x^{2}y\right]}{{\left(3y^2-x^2\right)}^2} \\ \frac{d^{2}y}{d{x}^2}=\frac{4x^{2}y+\left(3y^2-x^2\right)\left(2y\right)-\left[{\displaystyle \frac{12x^2{y}^3}{3y^2-x^2}-4x^{2}y}\right]}{{\left(3y^2-x^2\right)}^2} \\ =\frac{6x^{2}y+6y^3-2x^{2}y-{\displaystyle \frac{24x^2{y}^3}{3y^2-x^2}}+4x^{2}y}{{\left(3y^2-x^2\right)}^2} \\ \left(3,-1\right)\left(x,y\right) \\ {\left(\frac{d^{2}y}{d{x}^2}\right)}_{\left(3,-1\right)}=\frac{6{\left(3\right)}^2\left(-1\right)+6y^3-2{\left(3\right)}^2\left(1\right)-{\displaystyle \frac{24{\left(3\right)}^2{\left(-1\right)}^3}{3{\left(-1\right)}^2-{\left(3\right)}^2}}+4{\left(3\right)}^2\left(-1\right)}{{\left(3{\left(-1\right)}^2-{\left(3\right)}^2\right)}^2} \\ =\frac{-96}{36} \\ =\frac{-8}{3} \end{gathered}$$uncaught exception: Invalid chunk

in file: /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 68
#0 /var/www/html/integration/lib/php/Boot.class.php(769): com_wiris_plugin_impl_HttpImpl_1(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Invalid chunk') #1 /var/www/html/integration/lib/haxe/Http.class.php(532): _hx_lambda->execute('Invalid chunk') #2 /var/www/html/integration/lib/php/Boot.class.php(769): haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Invalid chunk') #3 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(30): _hx_lambda->execute('Invalid chunk') #4 /var/www/html/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Invalid chunk') #5 /var/www/html/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), Object(sys_net_Socket), NULL) #6 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(43): haxe_Http->request(true) #7 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(268): com_wiris_plugin_impl_HttpImpl->request(true) #8 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(307): com_wiris_plugin_impl_RenderImpl->showImage('38b3e86d0c53200...', NULL, Object(PhpParamsProvider)) #9 /var/www/html/integration/createimage.php(17): com_wiris_plugin_impl_RenderImpl->createImage('

$$\begin{gathered} =\frac{6x^{2}y+6y^3-2x^{2}y-{\displaystyle \frac{24x^2{y}^3}{3y^2-x^2}}+4x^{2}y}{{\left(3y^2-x^2\right)}^2} \\ \left(3,-1\right)\left(x,y\right) \\ {\left(\frac{d^{2}y}{d{x}^2}\right)}_{\left(3,-1\right)}=\frac{6{\left(3\right)}^2\left(-1\right)+6y^3-2{\left(3\right)}^2\left(1\right)-{\displaystyle \frac{24{\left(3\right)}^2{\left(-1\right)}^3}{3{\left(-1\right)}^2-{\left(3\right)}^2}}+4{\left(3\right)}^2\left(-1\right)}{{\left(3{\left(-1\right)}^2-{\left(3\right)}^2\right)}^2} \\ =\frac{-96}{36} \\ =\frac{-8}{3} \end{gathered}$$