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implicitdiff

differentiation of a function defined by an equation

 Calling Sequence implicitdiff(f, y, x) implicitdiff(f, y, x1,...,xk) implicitdiff({f1,...,fm}, {y1,...,yn}, u, x) implicitdiff({f1,...,fm}, {y1,...,yn}, u, x1,...,xk) implicitdiff({f1,...,fm}, {y1,...,yn}, {u1,...,ur}, x) implicitdiff({f1,...,fm}, {y1,...,yn}, {u1,...,ur}, x1,...,xk)

Parameters

 f, f1, ..., fm - algebraic expressions or equations y, y1, ..., yn - (variable) names or function of dependent variables u, u1, ..., ur - names (of dependent variables) x, x1, ..., xk - names (of derivative variables) notation= - optional notation directive

Description

 • 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 $\left\{\mathrm{D}\left(y\right)=-\frac{\left(1+2xy\right)}{\left(z+y\right)}$, $\mathrm{D}\left(z\right)=\frac{\left(-1+2xz\right)}{\left(z+y\right)}\right\}$. 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 $\frac{\mathrm{dy}}{\mathrm{dx}}$, $\frac{{\partial }}{{\partial }x}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y$ 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.

Examples

 > $f≔y=\frac{{x}^{2}}{z}$
 ${f}{≔}{y}{=}\frac{{{x}}^{{2}}}{{z}}$ (1)
 > $\mathrm{implicitdiff}\left(f,y,x\right)$
 $\frac{{2}{}{x}}{{z}}$ (2)
 > $\mathrm{implicitdiff}\left(f,y,z\right)$
 ${-}\frac{{{x}}^{{2}}}{{{z}}^{{2}}}$ (3)
 > $f≔{x}^{2}+{y}^{3}=1$
 ${f}{≔}{{y}}^{{3}}{+}{{x}}^{{2}}{=}{1}$ (4)
 > $\mathrm{implicitdiff}\left(f,y,x\right)$
 ${-}\frac{{2}{}{x}}{{3}{}{{y}}^{{2}}}$ (5)
 > $\mathrm{implicitdiff}\left(f,x,y\right)$
 ${-}\frac{{3}{}{{y}}^{{2}}}{{2}{}{x}}$ (6)
 > $\mathrm{implicitdiff}\left(f,y,z\right)$
 ${0}$ (7)
 > $\mathrm{implicitdiff}\left(f,y\left(x\right),x\right)$
 ${-}\frac{{2}{}{x}}{{3}{}{{y}}^{{2}}}$ (8)

If the expression notation is used, the variable names must match:

 > $\mathrm{implicitdiff}\left(f,y\left(a\right),x\right)$
 > $\mathrm{implicitdiff}\left(f,y,x,x\right)$
 ${-}\frac{{2}{}\left({3}{}{{y}}^{{3}}{+}{4}{}{{x}}^{{2}}\right)}{{9}{}{{y}}^{{5}}}$ (9)
 > $\mathrm{implicitdiff}\left(f,z,x\right)$
 ${\mathrm{FAIL}}$ (10)
 > $f≔a{x}^{3}y-\frac{2y}{z}={z}^{2}$
 ${f}{≔}{a}{}{{x}}^{{3}}{}{y}{-}\frac{{2}{}{y}}{{z}}{=}{{z}}^{{2}}$ (11)
 > $\mathrm{implicitdiff}\left(f,y\left(x,z\right),x\right)$
 ${-}\frac{{3}{}{a}{}{{x}}^{{2}}{}{y}{}{z}}{{a}{}{{x}}^{{3}}{}{z}{-}{2}}$ (12)
 > $\mathrm{implicitdiff}\left(f,y\left(x,z\right),x,z\right)$
 $\frac{{6}{}{a}{}{{x}}^{{2}}{}\left({-}{{z}}^{{3}}{+}{2}{}{y}\right)}{{\left({a}{}{{x}}^{{3}}{}{z}{-}{2}\right)}^{{2}}}$ (13)
 > $f≔{y}^{2}-2xz=1$
 ${f}{≔}{-}{2}{}{x}{}{z}{+}{{y}}^{{2}}{=}{1}$ (14)
 > $g≔{x}^{2}-\mathrm{exp}\left(xz\right)=y$
 ${g}{≔}{{x}}^{{2}}{-}{{ⅇ}}^{{x}{}{z}}{=}{y}$ (15)
 > $\mathrm{implicitdiff}\left(\left\{f,g\right\},\left\{y,z\right\},y,x\right)$
 $\frac{{2}{}{x}}{{{ⅇ}}^{{x}{}{z}}{}{y}{+}{1}}$ (16)
 > $\mathrm{implicitdiff}\left(\left\{f,g\right\},\left\{y,z\right\},\left\{y,z\right\},x\right)$
 $\left\{{\mathrm{D}}{}\left({y}\right){=}\frac{{2}{}{x}}{{{ⅇ}}^{{x}{}{z}}{}{y}{+}{1}}{,}{\mathrm{D}}{}\left({z}\right){=}{-}\frac{{z}{}{{ⅇ}}^{{x}{}{z}}{}{y}{-}{2}{}{x}{}{y}{+}{z}}{\left({{ⅇ}}^{{x}{}{z}}{}{y}{+}{1}\right){}{x}}\right\}$ (17)
 > $\mathrm{implicitdiff}\left(\left\{f,g\right\},\left\{y\left(x\right),z\left(x\right)\right\},\left\{y,z\right\},x,\mathrm{notation}=\mathrm{Diff}\right)$
 $\left\{\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{=}\frac{{2}{}{x}}{{{ⅇ}}^{{x}{}{z}}{}{y}{+}{1}}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{=}{-}\frac{{z}{}{{ⅇ}}^{{x}{}{z}}{}{y}{-}{2}{}{x}{}{y}{+}{z}}{\left({{ⅇ}}^{{x}{}{z}}{}{y}{+}{1}\right){}{x}}\right\}$ (18)
 > $f≔a\mathrm{sin}\left(uv\right)+b\mathrm{cos}\left(wx\right)=c$
 ${f}{≔}{a}{}{\mathrm{sin}}{}\left({u}{}{v}\right){+}{b}{}{\mathrm{cos}}{}\left({w}{}{x}\right){=}{c}$ (19)
 > $g≔u+v+w+x=z$
 ${g}{≔}{u}{+}{v}{+}{w}{+}{x}{=}{z}$ (20)
 > $h≔uv+wx=z$
 ${h}{≔}{u}{}{v}{+}{w}{}{x}{=}{z}$ (21)
 > $\mathrm{implicitdiff}\left(\left\{f,g,h\right\},\left\{u\left(x,z\right),v\left(x,z\right),w\left(x,z\right)\right\},u,z\right)$
 $\frac{{u}{}{\mathrm{cos}}{}\left({u}{}{v}\right){}{a}{}{x}{+}{u}{}{\mathrm{sin}}{}\left({w}{}{x}\right){}{b}{}{x}{-}{u}{}{\mathrm{cos}}{}\left({u}{}{v}\right){}{a}{-}{\mathrm{sin}}{}\left({w}{}{x}\right){}{b}{}{x}}{{x}{}\left({\mathrm{cos}}{}\left({u}{}{v}\right){}{a}{+}{\mathrm{sin}}{}\left({w}{}{x}\right){}{b}\right){}\left({-}{v}{+}{u}\right)}$ (22)
 > $\mathrm{implicitdiff}\left(\left\{g,h\right\},\left\{u\left(x,z\right),v\left(x,z\right),w\left(x,z\right)\right\},\left\{u,v,w\right\},z\right)$
 $\left\{{{\mathrm{D}}}_{{2}}{}\left({u}\right){=}{-}\frac{{x}{-}{1}}{{v}{-}{x}}{-}\frac{\left({u}{-}{x}\right){}{{\mathrm{D}}}_{{2}}{}\left({v}\right)}{{v}{-}{x}}{,}{{\mathrm{D}}}_{{2}}{}\left({v}\right){=}{{\mathrm{D}}}_{{2}}{}\left({v}\right){,}{{\mathrm{D}}}_{{2}}{}\left({w}\right){=}\frac{\left({-}{v}{+}{u}\right){}{{\mathrm{D}}}_{{2}}{}\left({v}\right)}{{v}{-}{x}}{+}\frac{{v}{-}{1}}{{v}{-}{x}}\right\}$ (23)
 > $\mathrm{implicitdiff}\left(\left\{f,g,h\right\},\left\{u\left(x,z\right),v\left(x,z\right)\right\},u,z\right)$
 ${\mathrm{FAIL}}$ (24)