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

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

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

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

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

A) 2x
B) x²
C) 1/x
D) 2

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

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

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

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

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

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

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

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

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

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

A) Prime ideals.
B) Maximal ideals.
C) Minimal 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) p
B) a_d
C) d
D) u_p

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

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

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

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

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 used only in polynomial algebra.
B) They are considered as belonging to differential algebra.
C) They serve as examples of non-commutative rings without derivations.
D) They are unrelated to differential 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) (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)) = p(y + a)
C) E^a(p(y)) = Mer(f(y), ∂y)
D) E^a(p(y)) = p(y) ⋅ ∂y

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

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