diff --git a/src/utils.jl b/src/utils.jl
index 4cc9c23..4dff349 100644
--- a/src/utils.jl
+++ b/src/utils.jl
@@ -1,32 +1,46 @@
 using ChainRules
 using ChainRulesCore
-using Symbolics: @variables
-using SymbolicUtils, SymbolicUtils.Code
-using SymbolicUtils: Pow
+using Symbolics: @variables, @rule, unwrap, isdiv
+using SymbolicUtils.Code: toexpr
 
-dummy = (NoTangent(), 1)
-@variables z
+"""
+Pick a strategy for raising the derivative of a function. If the derivative is like 1 over something, raise with the division rule; otherwise, raise with the multiplication rule.
+"""
+function get_term_raiser(func)
+    @variables z
+    r1 = @rule -1 * (1 / ~x) => (-1) / ~x
+    der = frule((NoTangent(), true), func, z)[2]
+    term = unwrap(der)
+    maybe_rewrite = r1(term)
+    if maybe_rewrite !== nothing
+        term = maybe_rewrite
+    end
+    if isdiv(term) && (term.num == 1 || term.num == -1)
+        term.den * term.num, raiseinv
+    else
+        term, raise
+    end
+end
 
 function define_unary_function(func, m)
     F = typeof(func)
-    # base case
+    # First order: call frule directly
     @eval m function (op::$F)(t::TaylorScalar{T, 1}) where {T}
         t0 = value(t)
         t1 = first(partials(t))
         f0, f1 = frule((NoTangent(), t1), op, t0)
         TaylorScalar{T, 1}(f0, zero_tangent(f0) + f1)
     end
-    der = frule(dummy, func, z)[2]
-    term, raiser = der isa Pow && der.exp == -1 ? (der.base, raiseinv) : (der, raise)
-    # recursion by raising
+    term, raiser = get_term_raiser(func)
+    # Higher order: recursion by raising
     @eval m @generated function (op::$F)(t::TaylorScalar{T, N}) where {T, N}
-        der_expr = $(QuoteNode(toexpr(term)))
+        expr = $(QuoteNode(toexpr(term)))
         f = $func
         quote
             $(Expr(:meta, :inline))
             z = TaylorScalar{T, N - 1}(t)
             f0 = $f(value(t)[1])
-            df = zero_tangent(z) + $der_expr
+            df = zero_tangent(z) + $expr
             $$raiser(f0, df, t)
         end
     end