Implicit Differentiation with Two Equations - Maple Programming Help

Implicit Differentiation with Two Equations

 Description Using implicit differentiation, compute the derivatives $\frac{\mathrm{dy}}{\mathrm{dx}}$ and $\frac{\mathrm{dz}}{\mathrm{dx}}$ implicitly from the equations $f\left(x,y,z\right)=g\left(x,y,z\right)$ and $F\left(x,y,z\right)=G\left(x,y,z\right)$.

Implicit Differentiation with Two Equations

Enter equations:

 >
 ${{y}}^{{2}}{-}{2}{}{x}{}{z}{=}{1}$ (1)
 >
 ${{x}}^{{2}}{-}{\mathrm{cos}}{}\left({x}{}{z}\right){=}{y}$ (2)

Obtain $\frac{\mathrm{dy}}{\mathrm{dx}}$:

 >
 ${-}\frac{{2}{}{x}}{{\mathrm{sin}}{}\left({x}{}{z}\right){}{y}{-}{1}}$ (3)

Obtain $\frac{\mathrm{dz}}{\mathrm{dx}}$:

 > $\mathrm{implicitdiff}\left(\left\{,\right\},\left\{{y}{,}{z}\right\},{z},{x}\right)$
 ${-}\frac{{2}{}{y}{}{x}{+}{z}{}{\mathrm{sin}}{}\left({x}{}{z}\right){}{y}{-}{z}}{\left({\mathrm{sin}}{}\left({x}{}{z}\right){}{y}{-}{1}\right){}{x}}$ (4)

Stepwise Calculation:

Replace $y$ with $y\left(x\right)$ and
$z$ with $z\left(x\right)$:

 >
 $\left[{{y}{}\left({x}\right)}^{{2}}{-}{2}{}{x}{}{z}{}\left({x}\right){=}{1}{,}{{x}}^{{2}}{-}{\mathrm{cos}}{}\left({x}{}{z}{}\left({x}\right)\right){=}{y}{}\left({x}\right)\right]$ (5)

Apply $\frac{d}{\mathrm{dx}}$:

 > $\mathrm{diff}\left(,{x}\right)$
 $\left[{2}{}{y}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){-}{2}{}{z}{}\left({x}\right){-}{2}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{z}{}\left({x}\right)\right){=}{0}{,}{2}{}{x}{+}{\mathrm{sin}}{}\left({x}{}{z}{}\left({x}\right)\right){}\left({z}{}\left({x}\right){+}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{z}{}\left({x}\right)\right)\right){=}\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right]$ (6)

Solve for both derivatives:

 >
 $\left[\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right){=}{-}\frac{{2}{}{x}}{{\mathrm{sin}}{}\left({x}{}{z}{}\left({x}\right)\right){}{y}{}\left({x}\right){-}{1}}{,}\frac{{ⅆ}}{{ⅆ}{x}}{}{z}{}\left({x}\right){=}{-}\frac{{2}{}{y}{}\left({x}\right){}{x}{+}{z}{}\left({x}\right){}{\mathrm{sin}}{}\left({x}{}{z}{}\left({x}\right)\right){}{y}{}\left({x}\right){-}{z}{}\left({x}\right)}{\left({\mathrm{sin}}{}\left({x}{}{z}{}\left({x}\right)\right){}{y}{}\left({x}\right){-}{1}\right){}{x}}\right]$ (7)

Replace $y\left(x\right)$ and $z\left(x\right)$ with $y$ and $z$, respectively:

 >
 $\left[\begin{array}{c}\frac{{\mathrm{dy}}}{{\mathrm{dx}}}\\ \frac{{\mathrm{dz}}}{{\mathrm{dx}}}\end{array}\right]{=}\left[\begin{array}{c}{-}\frac{{2}{}{x}}{{\mathrm{sin}}{}\left({x}{}{z}\right){}{y}{-}{1}}\\ {-}\frac{{2}{}{y}{}{x}{+}{z}{}{\mathrm{sin}}{}\left({x}{}{z}\right){}{y}{-}{z}}{\left({\mathrm{sin}}{}\left({x}{}{z}\right){}{y}{-}{1}\right){}{x}}\end{array}\right]$ (8)

 Commands Used