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

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

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

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

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

A) x²
B) 2
C) 2x
D) 1/x

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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