Standard functions - nth order derivatives

S.No f(x) f(n)(x) Condition
1.a (ax + b)m m(m – 1).......(m – n + 1) . an (ax + b)m – n where a, b, m are real numbers, a ≠ 0, (ax + b) > 0, ∀ n ∈ N
1.b (ax + b)m where a, b, m are real numbers, a ≠ 0, (ax + b) > 0, m > n, ∀ m, n ∈ N
1.c (ax + b)m 0 where a, b, m are real numbers, a ≠ 0, (ax + b) > 0, m < n, ∀ m, n ∈ N
1.d (ax + b)m n! an where a, b, m are real numbers, a ≠ 0, (ax + b) > 0, m = n, ∀ m, n ∈ N
2 where a, b are real numbers, a ≠ 0, (ax + b) > 0, ∀ n ∈ N
3 loge (ax + b) where a, b are real numbers, a ≠ 0, (ax + b) > 0, ∀ n ∈ N
4 e(ax + b) an.eax + b ∀ x ∈ R and ∀ n ∈ N
5 cax + b an (log c)n . cax + b c > 0, ∀ x ∈ R and ∀ n ∈ N
6 sin (ax + b) an sin(ax + b + nπ/2) ∀ x ∈ R and ∀ n ∈ N
7 cos (ax + b) an cos(ax + b + nπ/2) ∀ x ∈ R and ∀ n ∈ N
8 eax sin(bx + c) rn eax sin(bx + c + nθ) ∀ x ∈ R and ∀ n ∈ N, where r = ≠ 0, cos θ = a/r and sin θ = b/r
9 eax cos(bx + c) rn eax cos(bx + c + nθ) ∀ x ∈ R and ∀ n ∈ N, where r = ≠ 0, cos θ = a/r and sin θ = b/r