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

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

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

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

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

A) 2x
B) 2
C) 1/x
D) x²

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)