Ticket #12922: 16191.patch

sage/calculus/all.py

@@ -4 +4 @@
 
 from functional import (diff, derivative,
                         expand,
-                        taylor, simplify)
+                        taylor, simplify, implicit_derivative)
 
 from functions import (wronskian,jacobian)
 

sage/calculus/functional.py

@@ -32 +32 @@
 
 from calculus import SR
 from sage.symbolic.expression import Expression
+from sage.symbolic.function_factory import SymbolicFunction
 
 def simplify(f):
     r"""
@@ -433 +434 @@
         return x.expand(*args, **kwds)
     except AttributeError:
         return x
+
+
+def implicit_derivative(Y,X,F,n=1):
+    """
+    Computes the n'th derivative of Y with respect to X given implicitly by F.
+
+    EXAMPLES:
+
+    ::
+
+        sage: var('x,y')
+        (x, y)
+        sage: implicit_derivative(y,x,cos(x)*sin(y))
+        sin(x)*sin(y)/(cos(x)*cos(y))
+        sage: implicit_derivative(y,x,x*y^2,3)
+        -1/4*(y + 2*y/x)/x^2 + 1/4*(2*y^2/x - y^2/x^2)/(x*y) - 3/4*y/x^3
+    """
+    x=SR.symbol('x')
+    yy=SR.symbol('yy')
+    y=SymbolicFunction('y',1)(x)
+    f=SymbolicFunction('f',2)(x,yy)
+    Fx=f.diff(x)
+    Fy=f.diff(yy)
+    G=-(Fx/Fy)
+    G=G.subs_expr({yy:y})
+    di={y.diff(x):-F.diff(X)/F.diff(Y)}
+    R=G
+    S=G.diff(x,n-1)
+    for i in range(n+1):
+        di[y.diff(x,i+1)(x=x)]=R
+        S=S.subs_expr(di)
+        R=G.diff(x,i)
+        for j in range(n+1-i):
+            di[f.diff(x,i,yy,j)(x=x,yy=y)]=F.diff(X,i,Y,j)
+            S=S.subs_expr(di)
+    return S