The implicitdiff(f, y, x) (implicit differentiation) calling sequence computes $\frac{\mathrm{dy}}{\mathrm{dx}}$, the partial derivative of the function y with respect to x. The input f defines y as a function of x implicitly. It must be an equation in x and y or an algebraic expression, which is understood to be equated to zero.  For example, the call implicitdiff(x^2*y+y^2=1,y,x) computes the derivative of y with respect to x.  Here, y is implicitly a function of x. The result returned is $-\frac{2xy}{x^2+2y}$.

The second argument y specifies the dependent variables, the independent variables, and the constants.  If y is a name, this means that y is the dependent variable.  All other names, which appear in the input f and the derivative variable(s) x and are not of type constant, are treated as independent variables. For example, the call implicitdiff(R=P*V/T, P, T) specifies P, the dependent variable, is to be regarded as a function of R, P, and T the independent variables. If y is a function $y\left(\mathrm{x1},...,\mathrm{xj}\right)$, this states the independent variables and their order explicitly. All other variables appearing in the input f are implicitly understood to be constants. For example, the call implicitdiff(R=P*V/T, P(V, T), T) specifies that P is a function of T and V, and the variable R is a constant. The result is $\frac{P}{T}$.

Higher order partial derivatives are specified by giving more variables as optional arguments, exactly as with the diff command.

The implicitdiff routine will return the value FAIL if the derivative does not exist.  This would happen, for instance, if the first argument f is not a function of y.

The remaining four calling sequences specify the case of m equations $\mathrm{f1},...,\mathrm{fm}$ defining n functions $\mathrm{y1},...,\mathrm{yn}$ implicitly. The first argument $\mathrm{f1},...,\mathrm{fm}$ must be a set of equations or algebraic expressions which are understood to be equated to zero. The second argument $\mathrm{y1},...,\mathrm{yn}$ specifies the dependent variables, the independent variables and the constants as in the previous calling sequences. Note that if the equations $\mathrm{f1},...,\mathrm{fm}$ are overdetermined, the implicitdiff command may return FAIL.

The call implicitdiff({f1,...,fm},{y1,...,yn}, u, x) computes the derivative of the function u with respect to x where u must be one of the given y's. The call implicitdiff({f1,...,fm},{y1,...,yn}, u, x1,...,xk) computes higher order derivatives of u. For example, the call implicitdiff({x^2+y=z, x+y*z=1}, {y, z}, y, x) computes $\frac{\mathrm{dy}}{\mathrm{dx}}$.  The result is $-\frac{2xy+1}{z+y}$.

The call implicitdiff({f1,...,fm},{y1,...,yn},{u1,...,ur}, x) computes the partial derivatives of the functions $\mathrm{u1},...,\mathrm{ur}$ with respect to x.  For example, the call implicitdiff({x^2+y=z, x+y*z=1},{y,z},{y,z}, x) computes $\frac{\mathrm{dy}}{\mathrm{dx}}$ and $\frac{\mathrm{dz}}{\mathrm{dx}}$. The result is $\{\mathrm{D}\left(y\right)=-\frac{\left(1+2xy\right)}{\left(z+y\right)}$, $\mathrm{D}\left(z\right)=\frac{\left(-1+2xz\right)}{\left(z+y\right)}\}$. The result returned is the set of equations of the form $\frac{\mathrm{dy}}{\mathrm{dx}}=F\left(x\,y\,z\right)$.  The notation used to label the partial derivatives $\frac{\mathrm{dy}}{\mathrm{dx}}$ can be either Maple's D notation (the default) or a subscripted Diff notation. If the last argument is $\mathrm{notation}=\mathrm{D}$ or no notational directive is given, then Maple's $\mathrm{D}$ notation is used. For functions of one variable, $y\left(x\right)$, the notation $\mathrm{D}\left(y\right)$ will be used.  For functions of more than one variable, the ${\mathrm{D}}_i\left(y\right)$ notation will be used. If the Diff notation is specified, then instead of using $\mathrm{D}\left(y\right)$ for $\frac{\mathrm{dy}}{\mathrm{dx}}$, $\frac{\ⅆy}{\ⅆx}$ is used. And instead of using ${\mathrm{D}}_1\left(y\right)$ for $\frac{\mathrm{dy}}{\mathrm{dx}}$ where y is a function of more than one variable, say $y\left(x\,z\right)$ then Diff(y, x)[z] is used.

$f≔y=\frac{x^2}{z}$

$f≔y=\frac{x^2}{z}$

$\mathrm{implicitdiff}\left(f\,y\,x\right)$

$\frac{2x}{z}$

$\mathrm{implicitdiff}\left(f\,y\,z\right)$

$-\frac{x^2}{z^2}$

$f≔x^2+y^3=1$

$f≔y^3+x^2=1$

$\mathrm{implicitdiff}\left(f\,y\,x\right)$

$-\frac{2x}{3y^2}$

$\mathrm{implicitdiff}\left(f\,x\,y\right)$

$-\frac{3y^2}{2x}$

$\mathrm{implicitdiff}\left(f\,y\,z\right)$

$0$

$\mathrm{implicitdiff}\left(f\,y\left(x\right)\,x\right)$

$-\frac{2x}{3y^2}$

$\mathrm{implicitdiff}\left(f\,y\left(a\right)\,x\right)$

Error, (in implicitdiff) 2nd argument y(a) must be a function of x

$\mathrm{implicitdiff}\left(f\,y\,x\,x\right)$

$-\frac{2\left(3y^3+4x^2\right)}{9y^5}$

$\mathrm{implicitdiff}\left(f\,z\,x\right)$

$\mathrm{FAIL}$

$f≔a{x}^{3}y-\frac{2y}{z}=z^2$

$f≔a{x}^{3}y-\frac{2y}{z}=z^2$

$\mathrm{implicitdiff}\left(f\,y\left(x\,z\right)\,x\right)$

$-\frac{3a{x}^{2}yz}{a{x}^{3}z-2}$

$\mathrm{implicitdiff}\left(f\,y\left(x\,z\right)\,x\,z\right)$

$\frac{6a{x}^2\left(-z^3+2y\right)}{{\left(a{x}^{3}z-2\right)}^2}$

$f≔y^2-2xz=1$

$f≔-2xz+y^2=1$

$g≔x^2-\exp \left(xz\right)=y$

$g≔x^2-{\ⅇ}^{xz}=y$

$\mathrm{implicitdiff}\left(\left\{f\,g\right\}\,\left\{y\,z\right\}\,y\,x\right)$

$\frac{2x}{{\ⅇ}^{xz}y+1}$

$\mathrm{implicitdiff}\left(\left\{f\,g\right\}\,\left\{y\,z\right\}\,\left\{y\,z\right\}\,x\right)$

$\left\{\mathrm{D}\left(y\right)=\frac{2x}{{\ⅇ}^{xz}y+1}\,\mathrm{D}\left(z\right)=-\frac{z{\ⅇ}^{xz}y-2yx+z}{\left({\ⅇ}^{xz}y+1\right)x}\right\}$

$\mathrm{implicitdiff}\left(\left\{f\,g\right\}\,\left\{y\left(x\right)\,z\left(x\right)\right\}\,\left\{y\,z\right\}\,x\,\mathrm{notation}=\mathrm{Diff}\right)$

$\left\{\frac{\ⅆy}{\ⅆx}=\frac{2x}{{\ⅇ}^{xz}y+1}\,\frac{\ⅆz}{\ⅆx}=-\frac{z{\ⅇ}^{xz}y-2yx+z}{\left({\ⅇ}^{xz}y+1\right)x}\right\}$

$f≔a\sin \left(uv\right)+b\cos \left(wx\right)=c$

$f≔a\sin \left(uv\right)+b\cos \left(wx\right)=c$

$g≔u+v+w+x=z$

$g≔u+v+w+x=z$

$h≔uv+wx=z$

$h≔uv+wx=z$

$\mathrm{implicitdiff}\left(\left\{f\,g\,h\right\}\,\left\{u\left(x\,z\right)\,v\left(x\,z\right)\,w\left(x\,z\right)\right\}\,u\,z\right)$

$\frac{u\cos \left(uv\right)ax+u\sin \left(wx\right)bx-u\cos \left(uv\right)a-\sin \left(wx\right)bx}{x\left(\cos \left(uv\right)a+\sin \left(wx\right)b\right)\left(-v+u\right)}$

$\mathrm{implicitdiff}\left(\left\{g\,h\right\}\,\left\{u\left(x\,z\right)\,v\left(x\,z\right)\,w\left(x\,z\right)\right\}\,\left\{u\,v\,w\right\}\,z\right)$

$\left\{{\mathrm{D}}_2\left(u\right)=-\frac{-1+x}{v-x}-\frac{\left(u-x\right){\mathrm{D}}_2\left(v\right)}{v-x}\,{\mathrm{D}}_2\left(v\right)={\mathrm{D}}_2\left(v\right)\,{\mathrm{D}}_2\left(w\right)=\frac{\left(-v+u\right){\mathrm{D}}_2\left(v\right)}{v-x}+\frac{v-1}{v-x}\right\}$

$\mathrm{implicitdiff}\left(\left\{f\,g\,h\right\}\,\left\{u\left(x\,z\right)\,v\left(x\,z\right)\right\}\,u\,z\right)$

$\mathrm{FAIL}$