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implicitdiff

differentiation of a function defined by an equation

 

Calling Sequence

Parameters

Description

Examples

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=<derivative>

-

optional notation directive

Description

• 

The implicitdiff(f, y, x) (implicit differentiation) calling sequence computes dydx, 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 2xyx2&plus;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 yx1,...,xj, 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 PT.

• 

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 f1,...,fm defining n functions y1,...,yn implicitly. The first argument f1,...,fm must be a set of equations or algebraic expressions which are understood to be equated to zero. The second argument y1,...,yn specifies the dependent variables, the independent variables and the constants as in the previous calling sequences. Note that if the equations f1,...,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 dydx.  The result is 2xy&plus;1z&plus;y.

• 

The call implicitdiff({f1,...,fm},{y1,...,yn},{u1,...,ur}, x) computes the partial derivatives of the functions u1,...,ur with respect to x.  For example, the call implicitdiff({x^2+y=z, x+y*z=1},{y,z},{y,z}, x) computes dydx and dzdx. The result is {Dy=1+2xyz+y, Dz=1+2xzz+y}. The result returned is the set of equations of the form dydx&equals;Fx&comma;y&comma;z.  The notation used to label the partial derivatives dydx can be either Maple's D notation (the default) or a subscripted Diff notation. If the last argument is notation&equals;D or no notational directive is given, then Maple's D notation is used. For functions of one variable, yx, the notation Dy will be used.  For functions of more than one variable, the D&lsqb;i&rsqb;y notation will be used. If the Diff notation is specified, then instead of using Dy for dydx, xy is used. And instead of using D&lsqb;1&rsqb;y for dydx where y is a function of more than one variable, say yx&comma;z then Diff(y, x)[z] is used.

Examples

fy&equals;x2z

f:=y&equals;x2z

(1)

implicitdifff&comma;y&comma;x

2xz

(2)

implicitdifff&comma;y&comma;z

x2z2

(3)

fx2&plus;y3&equals;1

f:=y3&plus;x2&equals;1

(4)

implicitdifff&comma;y&comma;x

23xy2

(5)

implicitdifff&comma;x&comma;y

32y2x

(6)

implicitdifff&comma;y&comma;z

0

(7)

implicitdifff&comma;yx&comma;x

23xy2

(8)

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

implicitdifff&comma;ya&comma;x

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

implicitdifff&comma;y&comma;x&comma;x

293y3&plus;4x2y5

(9)

implicitdifff&comma;z&comma;x

FAIL

(10)

fax3y2yz&equals;z2

f:=ax3y2yz&equals;z2

(11)

implicitdifff&comma;yx&comma;z&comma;x

3ax2yzax3z2

(12)

implicitdifff&comma;yx&comma;z&comma;x&comma;z

6ax2z3&plus;2yax3z22

(13)

fy22xz&equals;1

f:=2xz&plus;y2&equals;1

(14)

gx2&ExponentialE;xz&equals;y

g:=x2&ExponentialE;xz&equals;y

(15)

implicitdifff&comma;g&comma;y&comma;z&comma;y&comma;x

2x&ExponentialE;xzy&plus;1

(16)

implicitdifff&comma;g&comma;y&comma;z&comma;y&comma;z&comma;x

Dy&equals;2x&ExponentialE;xzy&plus;1&comma;Dz&equals;z&ExponentialE;xzy2xy&plus;z&ExponentialE;xzy&plus;1x

(17)

implicitdifff&comma;g&comma;yx&comma;zx&comma;y&comma;z&comma;x&comma;notation&equals;Diff

xy&equals;2x&ExponentialE;xzy&plus;1&comma;xz&equals;z&ExponentialE;xzy2xy&plus;z&ExponentialE;xzy&plus;1x

(18)

fasinuv&plus;bcoswx&equals;c

f:=asinuv&plus;bcoswx&equals;c

(19)

gu&plus;v&plus;w&plus;x&equals;z

g:=u&plus;v&plus;w&plus;x&equals;z

(20)

huv&plus;wx&equals;z

h:=uv&plus;wx&equals;z

(21)

implicitdifff&comma;g&comma;h&comma;ux&comma;z&comma;vx&comma;z&comma;wx&comma;z&comma;u&comma;z

ucosuvax&plus;usinwxbxucosuvasinwxbxxacosuv&plus;sinwxbvu

(22)

implicitdiffg&comma;h&comma;ux&comma;z&comma;vx&comma;z&comma;wx&comma;z&comma;u&comma;v&comma;w&comma;z

D2u&equals;x1vxx&plus;uD2vvx&comma;D2v&equals;D2v&comma;D2w&equals;vuD2vvx&plus;v1vx

(23)

implicitdifff&comma;g&comma;h&comma;ux&comma;z&comma;vx&comma;z&comma;u&comma;z

FAIL

(24)

See Also

D

diff

 


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