diff -r dccf609cd020 -r a5ee4bfb0b87 src/mapping.rs
--- a/src/mapping.rs	Mon Dec 08 15:23:42 2025 -0500
+++ b/src/mapping.rs	Thu Jan 15 16:12:47 2026 -0500
@@ -83,6 +83,14 @@
     /// Calculate differential at `x`
     fn differential<I: Instance<Domain>>(&self, x: I) -> Self::DerivativeDomain;
 
+    /// Calculate differential and value at `x`
+    fn apply_and_differential<I: Instance<Domain>>(
+        &self,
+        x: I,
+    ) -> (Self::Codomain, Self::DerivativeDomain) {
+        x.eval_ref(|xo| (self.apply(xo), self.differential(xo)))
+    }
+
     /// Form the differential mapping of `self`.
     fn diff(self) -> Self::Differential<'static>;
 
@@ -108,6 +116,16 @@
 
     /// Compute the differential of `self` at `x`, consuming the input.
     fn differential_impl<I: Instance<X>>(&self, x: I) -> Self::Derivative;
+
+    fn apply_and_differential_impl<I: Instance<X>>(
+        &self,
+        x: I,
+    ) -> (Self::Codomain, Self::Derivative)
+    where
+        Self: Mapping<X>,
+    {
+        x.eval_ref(|xo| (self.apply(xo), self.differential_impl(xo)))
+    }
 }
 
 impl<T, Domain> DifferentiableMapping<Domain> for T
@@ -126,6 +144,14 @@
         self.differential_impl(x)
     }
 
+    #[inline]
+    fn apply_and_differential<I: Instance<Domain>>(
+        &self,
+        x: I,
+    ) -> (T::Codomain, Self::DerivativeDomain) {
+        self.apply_and_differential_impl(x)
+    }
+
     fn diff(self) -> Differential<'static, Domain, Self> {
         Differential { g: MyCow::Owned(self), _space: PhantomData }
     }