diff --git a/sumpy/kernel.py b/sumpy/kernel.py
index 26f27adc800ac23fe1e64f27b740ac0f8ecd2466..039368916020ccfc4cb6f5a4f20bb92be62a5fa1 100644
--- a/sumpy/kernel.py
+++ b/sumpy/kernel.py
@@ -1030,12 +1030,15 @@ class AxisTargetDerivative(DerivativeBase):
 
         target_vec = make_sympy_vector(self.target_array_name, self.dim)
 
+        # bvec = tgt - ctr
         expr = self.inner_kernel.postprocess_at_target(expr, bvec)
         if isinstance(expr, DifferentiatedExprDerivativeTaker):
             transformation = diff_transformation(expr.derivative_transformation,
                     self.axis, target_vec)
             return DifferentiatedExprDerivativeTaker(expr.taker, transformation)
         else:
+            # Since `bvec` and `tgt` are two different symbolic variables
+            # need to differentiate by both to get the correct answer
             return expr.diff(bvec[self.axis]) + expr.diff(target_vec[self.axis])
 
     def replace_base_kernel(self, new_base_kernel):
@@ -1129,6 +1132,8 @@ class DirectionalTargetDerivative(DirectionalDerivative):
         if not isinstance(expr, DifferentiatedExprDerivativeTaker):
             result = 0
             for axis in range(self.dim):
+                # Since `bvec` and `tgt` are two different symbolic variables
+                # need to differentiate by both to get the correct answer
                 result += (expr.diff(bvec[axis]) + expr.diff(target_vec[axis])) \
                         * dir_vec[axis]
             return result
diff --git a/sumpy/tools.py b/sumpy/tools.py
index df7080537c253509d998486ac8b8cb3e6e09f9eb..d37ef2aa8f98337febebf20320cd91aaec78f6fa 100644
--- a/sumpy/tools.py
+++ b/sumpy/tools.py
@@ -433,11 +433,19 @@ class DifferentiatedExprDerivativeTaker:
 
 
 def diff_transformation(derivative_transformation, variable_idx, variables):
+    """Differentiate a derivative transformation dictionary using the
+    variable given by **variable_idx** and return a new derivative transformation
+    dictionary
+    """
     new_transformation = defaultdict(lambda: 0)
     for mi, coeff in derivative_transformation.items():
+        # In the case where we have x * u.diff(x), the result should
+        # be x.diff(x) + x * u.diff(x, x)
+        # Calculate the first term by differentiating the coefficients
         new_coeff = sym.sympify(coeff).diff(variables[variable_idx])
         if new_coeff != 0:
             new_transformation[mi] += new_coeff
+        # Next calculate the second term by differentitating the derivatives
         new_mi = list(mi)
         new_mi[variable_idx] += 1
         new_transformation[tuple(new_mi)] += coeff