Implicit differentiation

13.2.3 Implicit differentiation

The implicitdiff command can differentiate implicitly defined functions or expressions containing implicitly defined functions. It has three different calling sequences.

To implicitly differentiate dependent variables, implicitdiff takes four arguments:
- constraints, an equation or list of equations which implicitly define the dependent variables as functions of the independent variables; these will be of the form
  g_i(x₁,…,x_n,y₁,…,y_m)=0
  
  for i=1,2,…,m, where x₁,…,x_n are the independent variables and y₁,…,y_m are the dependent variables.
- depvars, the list of dependent variables, where each dependent variable can optionally be written as a function of the x_i or the name written as a function of the independent variables y_i(x₁,…,x_n). If there is only one dependent variable, this can be omitted.
- y, a dependent variable or a list of dependent variables to be differentiated.
- diffvars, a sequence of independent variables x_i₁,…,x_{i_k} with respect to differentiate.
implicitdiff(constraints ⟨,depvars ⟩,y,diffvars) returns the derivative (or list of derivatives) of y with respect to diffvars.
To find a specified derivative of an expression containing implicitly defined functions, implicitdiff takes four arguments:
- expr, a differentiable expression involving independent variables x₁,x₂,…,x_n and dependent variables y₁,y₂,…,y_m.
- constraints, an equation or list of equations which implicitly define the dependent variables as functions of the independent variables; these will be of the form
  g_i(x₁,…,x_n,y₁,…,y_m)=0
  
  for i=1,2,…,m.
- depvars, the dependent variable or list of dependent variables, where each dependent variable can either be the variable name y_i or the name written as a function of the independent variables y_i(x₁,…,x_n)).
- diffvars, a sequence of independent variables x_i₁,…,x_{i_k} with respect to which expr is differentiated.
implicitdiff(expr,implicitdef,depvars,diffvars) returns the expression expr differentiated with respect to diffvars.
To find all kth order derivatives of an expression involving implicitly defined functions, implicitdiff takes four mandatory arguments and one optional argument:
- expr, a differentiable expression involving independent variables x₁,x₂,…,x_n and dependent variables y₁,y₂,…,y_m.
- constraints, an equation or list of equations which implicitly define the dependent variables as functions of the independent variables; these will be of the form
  g_i(x₁,…,x_n,y₁,…,y_m)=0
  
  for i=1,2,…,m.
- vars, a list [x₁,…,x_n,y₁,…,y_m] of the independent and dependent variables entered as symbols in single list such that dependent variables come last.
- order=k, where k is the order of the derivatives to be taken.
- Optionally, a, a point where the partial derivatives should be evaluated at.
implicitdiff(expr,implicitdef,vars,order=k ⟨,a ⟩) returns all partial derivatives of order k. If k=1 they are returned in a single list, which represents the gradient of expr with respect to independent variables. If k=2 the corresponding Hessian matrix is returned (see Section 13.7.3). If k>2, a table with keys in form [k₁,k₂,…,k_n], where ∑_i=1ⁿk_i=k, is returned. Such a key corresponds to
∂^k f

∂ var₁^k₁ ∂ var₂^k₂ ⋯ ∂ var_n^k_n

.

Examples

implicitdiff(x^2*y+y^2=1,y,x)

−

2 x y

x²+2 y

implicitdiff([x^2+y=z,x+y*z=1],[y(x),z(x)],y,x)

−2 x y−1

y+z

implicitdiff(x*y,-2x^3+15x^2*y+11y^3-24y=0,y(x),x)

2 x³−5 x² y+11 y³−8 y

5 x²+11 y²−8

f:=x*y*z:; g:=-2x^3+15x^2*y+11y^3-24y=0:; implicitdiff(f,g,[x,z,y],order=1)

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2 x³ z−5 x² y z+11 y³ z−8 y z

5 x²+11 y²−8

,x y

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implicitdiff(f,g,[x,z,y],order=2,[1,-1,0])

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In the following example, the value of ∂⁴ f/∂ x⁴ is computed at the point (x=0,y=0,z).

pd:=implicitdiff(f,g,[x,z,y],order=4,[0,z,0]):; pd[4,0]

−2 z