diff --git a/src/dual.jl b/src/dual.jl
index ca3a2cbe..7e8ec110 100644
--- a/src/dual.jl
+++ b/src/dual.jl
@@ -552,7 +552,7 @@ end
 # exponentiation #
 #----------------#
 
-for f in (:(Base.:^), :(NaNMath.pow))
+for (f, log) in ((:(Base.:^), :(Base.log)), (:(NaNMath.pow), :(NaNMath.log)))
     @eval begin
         @define_binary_dual_op(
             $f,
@@ -565,7 +565,7 @@ for f in (:(Base.:^), :(NaNMath.pow))
                 elseif iszero(vx) && vy > 0
                     logval = zero(vx)
                 else
-                    logval = expv * log(vx)
+                    logval = expv * ($log)(vx)
                 end
                 new_partials = _mul_partials(partials(x), partials(y), powval, logval)
                 return Dual{Txy}(expv, new_partials)
@@ -583,7 +583,7 @@ for f in (:(Base.:^), :(NaNMath.pow))
             begin
                 v = value(y)
                 expv = ($f)(x, v)
-                deriv = (iszero(x) && v > 0) ? zero(expv) : expv*log(x)
+                deriv = (iszero(x) && v > 0) ? zero(expv) : expv*($log)(x)
                 return Dual{Ty}(expv, deriv * partials(y))
             end
         )
diff --git a/test/DerivativeTest.jl b/test/DerivativeTest.jl
index 4b7463c8..4de1a6de 100644
--- a/test/DerivativeTest.jl
+++ b/test/DerivativeTest.jl
@@ -1,6 +1,7 @@
 module DerivativeTest
 
 import Calculus
+import NaNMath
 
 using Test
 using Random
@@ -93,6 +94,17 @@ end
     @test (x -> ForwardDiff.derivative(y -> x^y,  1.5))(0.0) === 0.0
 end
 
+@testset "exponentiation with NaNMath" begin
+    @test isnan(ForwardDiff.derivative(x -> NaNMath.pow(NaN, x), 1.0))
+    @test isnan(ForwardDiff.derivative(x -> NaNMath.pow(x,NaN), 1.0))
+    @test !isnan(ForwardDiff.derivative(x -> NaNMath.pow(1.0, x),1.0))
+    @test isnan(ForwardDiff.derivative(x -> NaNMath.pow(x,0.5), -1.0))
+
+    @test isnan(ForwardDiff.derivative(x -> x^NaN, 2.0))
+    @test ForwardDiff.derivative(x -> x^2.0,2.0) == 4.0
+    @test_throws DomainError ForwardDiff.derivative(x -> x^0.5, -1.0)
+end
+
 @testset "dimension error for derivative" begin
     @test_throws DimensionMismatch ForwardDiff.derivative(sum, fill(2pi, 3))
 end
diff --git a/test/GradientTest.jl b/test/GradientTest.jl
index 5adfc8c7..a386c479 100644
--- a/test/GradientTest.jl
+++ b/test/GradientTest.jl
@@ -1,6 +1,7 @@
 module GradientTest
 
 import Calculus
+import NaNMath
 
 using Test
 using LinearAlgebra
@@ -200,6 +201,16 @@ end
     @test ForwardDiff.gradient(L -> logdet(L), Matrix(L)) ≈ [1.0 -1.3333333333333337; 0.0 1.666666666666667]
 end
 
+@testset "gradient for exponential with NaNMath" begin
+    @test isnan(ForwardDiff.gradient(x -> NaNMath.pow(x[1],x[1]), [NaN, 1.0])[1])
+    @test ForwardDiff.gradient(x -> NaNMath.pow(x[1], x[2]), [1.0, 1.0]) == [1.0, 0.0]
+    @test isnan(ForwardDiff.gradient((x) -> NaNMath.pow(x[1], x[2]), [-1.0, 0.5])[1])
+
+    @test isnan(ForwardDiff.gradient(x -> x[1]^x[2], [NaN, 1.0])[1])
+    @test ForwardDiff.gradient(x -> x[1]^x[2], [1.0, 1.0]) == [1.0, 0.0]
+    @test_throws DomainError ForwardDiff.gradient(x -> x[1]^x[2], [-1.0, 0.5])
+end
+
 @testset "branches in mul!" begin
     a, b = rand(3,3), rand(3,3)