A) Exponentiation
B) Derivative
C) Integration
D) Matrix multiplication

A) Quotient Rule
B) Chain Rule
C) Power Rule
D) Product Rule

A) The function itself
B) Pi
C) Zero
D) Infinity

A) tan(x)
B) cos(x)
C) csc(x)
D) -sin(x)

A) The function itself
B) Rate of change of the rate of change
C) A linear transformation
D) Average value of a function

A) 2x
B) 2
C) 1/x
D) x²

A) Composition
B) Addition
C) Multiplication
D) Differentiation

A) Chain Rule
B) Product Rule
C) Power Rule
D) Quotient Rule

A) Domain
B) Rate of change
C) Roots
D) Integral

A) Niels Henrik Abel
B) Joseph Ritt
C) David Hilbert
D) Ellis Kolchin

A) A non-commutative ring with no derivations.
B) A commutative ring equipped with one or more derivations that commute pairwise.
C) A set of all possible differentials in calculus.
D) A field without any derivation.

A) A commutative ring with no derivations.
B) A set of all possible differentials in calculus.
C) A differential ring that is also a field.
D) A non-commutative algebraic structure.

A) δ(cr) = δ(c)r
B) δ(cr) = cδ(r)
C) δ(cr) = crδ(c)
D) δ(cr) = rδ(c)

A) Generally, no.
B) Yes, always.
C) Only if S is infinite.
D) If S contains only constants.

A) HA ⊇ HΩ
B) HΩ ⊂ HA
C) HΩ ⊇ HA
D) HΩ = HA

A) (C .δ)
B) (Z .δ)
C) (R .δ)
D) (Q .δ)

A) Radical ideals.
B) Maximal ideals.
C) Prime ideals.
D) Minimal ideals.

A) Solving differential equations without any simplification.
B) Graph plotting of differential equations.
C) Numerical integration of differential equations.
D) Ranking derivatives, polynomials, and polynomial sets.

A) p
B) u_p
C) a_d
D) d

A) E^a ∘ T = T ∘ E^a
B) E^a ∘ T ≠ T ∘ E^a
C) E^a(p(y)) = p(y + a)
D) T' = T ∘ y - y ∘ T

A) Linear differential operator
B) Pincherle derivative
C) Differential meromorphic function field
D) Shift operator

A) δ(rⁿ) = rⁿδ(r)
B) δ(rⁿ) = nδ(r)r^n-1
C) δ(rⁿ) = δ(r)/r
D) δ(rⁿ) = nr^n-1δ(r)

A) The rank u_p^d
B) The leading coefficient a_d
C) The separant S_p
D) The constant term a₀

A) A total order and an admissible order defined by specific conditions.
B) Assigning equal rank to all derivatives.
C) Random assignment of ranks to derivatives.
D) Ignoring the order of derivatives.

A) They are unrelated to differential algebra.
B) They are considered as belonging to differential algebra.
C) They serve as examples of non-commutative rings without derivations.
D) They are used only in polynomial algebra.

A) δ(r/u) = (rδ(u) - δ(r))/u
B) δ(r/u) = u(δ(r) - rδ(u))
C) δ(r/u) = δ(r)/δ(u)
D) δ(r/u) = (δ(r)u - rδ(u))/u²

A) (Mer(f(y), ∂y))
B) (C{y}, p(y) ⋅ ∂y)
C) (E^a(p(y)) = p(y + a))
D) (T' = T ∘ y - y ∘ T)

A) E^a(p(y)) = T ∘ y - y ∘ T
B) E^a(p(y)) = Mer(f(y), ∂y)
C) E^a(p(y)) = p(y + a)
D) E^a(p(y)) = p(y) ⋅ ∂y

A) A set of all possible differentials in calculus.
B) A differential ring that contains K as a subring with matching derivations.
C) An algebraic structure unrelated to fields or rings.
D) A commutative ring without any derivation.

A) δ(u₁^e₁ ... u_n^e_n)/(u₁^e₁ ... u_n^e_n) = δ(u₁)/u₁ + ... + δ(u_n)/u_n
B) δ(u₁^e₁ ... u_n^e_n) = e₁(δ(u₁)) + ... + e_n(δ(u_n))
C) δ(u₁^e₁ ... u_n^e_n) = (u₁^e₁ ... u_n^e_n)(e₁δ(u₁) + ... + e_nδ(u_n))
D) δ(u₁^e₁ ... u_n^e_n)/(u₁^e₁ ... u_n^e_n) = e₁(δ(u₁)/u₁) + ... + e_n(δ(u_n)/u_n)