diff --git a/Project.toml b/Project.toml
index fc3740d..eade314 100644
--- a/Project.toml
+++ b/Project.toml
@@ -16,6 +16,7 @@ Symbolics = "0c5d862f-8b57-4792-8d23-62f2024744c7"
 UnPack = "3a884ed6-31ef-47d7-9d2a-63182c4928ed"
 
 [compat]
+ForwardDiff = "0.10.36"
 InvertedIndices = "1.3"
 QuadGK = "2.9"
 SpecialFunctions = "2.3"
diff --git a/test/Derivative.jl b/test/Derivative.jl
index 535e09a..417197c 100644
--- a/test/Derivative.jl
+++ b/test/Derivative.jl
@@ -6,15 +6,15 @@ half_derivative_of_x_square_minus_1_at_1 = 5/(3*sqrt(π))
 half_derivative_of_exp_at_1 = ℯ*erf(1)+1/sqrt(π)
 
 @testset "Test Caputo Direct Fractional Derivative" begin
-    @test isapprox(fracdiff(x->x, 0.5, 0, 1, 1e-8, CaputoDirect())[1], 2/sqrt(pi); atol = 1e-5)
-    @test isapprox(fracdiff(x->x^5, 0.5, 0, 3.2, 1e-8, CaputoDirect())[1], 4.300306216488329e2; atol = 1e-5)
-    @test isapprox(fracdiff(x->x^5, 0.25, 0, 3.2, 1e-8, CaputoDirect())[1], 382.1228113951680; atol = 1e-5)
-    @test isapprox(fracdiff(x->(x-3)^3+(x+2)^2+5, 0.5, 0, 0.83, 1e-8, CaputoDirect())[1], 23.899930581118699; atol = 1e-5)
-    @test isapprox(fracdiff(x->(x-3)^3+(x+2)^2+5, 0.25, 0, 0.83, 1e-8, CaputoDirect())[1], 22.96354626572943; atol = 1e-5)
-    @test isapprox(fracdiff(x->cos(x), 0.5, 0, 0.34, 1e-8, CaputoDirect())[1], -0.147174774607683; atol = 1e-5)
-    @test isapprox(fracdiff(x->cos(x), 0.25, 0, 0.34, 1e-8, CaputoDirect())[1], -0.093074344902431; atol = 1e-5)
-    @test isapprox(fracdiff(x->exp(-x^2), 0.5, 0, 2.3, 1e-8, CaputoDirect())[1], -0.505891017154289; atol = 1e-5)
-    @test isapprox(fracdiff(x->exp(-x^2), 0.25, 0, 2.3, 1e-8, CaputoDirect())[1], -0.762927553656252; atol = 1e-5)
+    @test isapprox(fracdiff(x->x, 0.5, 0, 1, 1e-4, CaputoDirect())[1], 2/sqrt(pi); atol = 1e-5)
+    @test isapprox(fracdiff(x->x^5, 0.5, 0, 3.2, 1e-4, CaputoDirect())[1], 4.300306216488329e2; atol = 1e-5)
+    @test isapprox(fracdiff(x->x^5, 0.25, 0, 3.2, 1e-4, CaputoDirect())[1], 382.1228113951680; atol = 1e-5)
+    @test isapprox(fracdiff(x->(x-3)^3+(x+2)^2+5, 0.5, 0, 0.83, 1e-4, CaputoDirect())[1], 23.899930581118699; atol = 1e-5)
+    @test isapprox(fracdiff(x->(x-3)^3+(x+2)^2+5, 0.25, 0, 0.83, 1e-4, CaputoDirect())[1], 22.96354626572943; atol = 1e-5)
+    @test isapprox(fracdiff(x->cos(x), 0.5, 0, 0.34, 1e-4, CaputoDirect())[1], -0.147174774607683; atol = 1e-5)
+    @test isapprox(fracdiff(x->cos(x), 0.25, 0, 0.34, 1e-4, CaputoDirect())[1], -0.093074344902431; atol = 1e-5)
+    @test isapprox(fracdiff(x->exp(-x^2), 0.5, 0, 2.3, 1e-4, CaputoDirect())[1], -0.505891017154289; atol = 1e-5)
+    @test isapprox(fracdiff(x->exp(-x^2), 0.25, 0, 2.3, 1e-4, CaputoDirect())[1], -0.762927553656252; atol = 1e-5)
 
     @test isapprox(fracdiff(x->x, 0.5, 0, [1, 2, 3], 0.0001, CaputoDirect()), [2/sqrt(pi), 1.5957691216057408, 1.9544100476116828]; atol=1e-3)
 end
diff --git a/test/Integral.jl b/test/Integral.jl
index 374dd70..95aa6e9 100644
--- a/test/Integral.jl
+++ b/test/Integral.jl
@@ -2,12 +2,12 @@ using FractionalCalculus
 using Test
 
 @testset "Test Direct Fractional Integral" begin
-    @test isapprox(fracint(x->x^4, 0.5, 0, 1.23, 1e-8, RLDirect())[1], 1.163931646862977; atol = 0.00001)
-    @test isapprox(fracint(x->x^4, 0.25, 0, 1.23, 1e-8, RLDirect())[1], 1.64294134724409; atol = 0.00001)
-    @test isapprox(fracint(x->x^4, 0.15, 0, 1.23, 1e-8, RLDirect())[1], 1.878978485456182; atol = 0.00001)
-    @test isapprox(fracint(x->sin(x), 0.5, 0, 0.97, 1e-8, RLDirect())[1], 0.6442601778843754; atol = 0.00001)
-    @test isapprox(fracint(x->sin(x), 0.25, 0, 0.97, 1e-8, RLDirect())[1], 0.7448298999485293; atol = 0.00001)
-    @test isapprox(fracint(x->sin(x), 0.15, 0, 0.97, 1e-8, RLDirect())[1], 0.7802555196986999; atol = 0.00001)
+    @test isapprox(fracint(x->x^4, 0.5, 0, 1.23, 1e-4, RLDirect())[1], 1.163931646862977; atol = 0.00001)
+    @test isapprox(fracint(x->x^4, 0.25, 0, 1.23, 1e-4, RLDirect())[1], 1.64294134724409; atol = 0.00001)
+    @test isapprox(fracint(x->x^4, 0.15, 0, 1.23, 1e-4, RLDirect())[1], 1.878978485456182; atol = 0.00001)
+    @test isapprox(fracint(x->sin(x), 0.5, 0, 0.97, 1e-4, RLDirect())[1], 0.6442601778843754; atol = 0.00001)
+    @test isapprox(fracint(x->sin(x), 0.25, 0, 0.97, 1e-4, RLDirect())[1], 0.7448298999485293; atol = 0.00001)
+    @test isapprox(fracint(x->sin(x), 0.15, 0, 0.97, 1e-4, RLDirect())[1], 0.7802555196986999; atol = 0.00001)
 end
 
 @testset "Test Piecewise Fractional Integral" begin