Theorems · Theorem · real analysis
deriv_fun_mul
∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {x : 𝕜} {𝔸 : Type u_3} [inst_1 : NormedRing 𝔸]
[inst_2 : NormedAlgebra 𝕜 𝔸] {c d : 𝕜 → 𝔸},
DifferentiableAt 𝕜 c x → DifferentiableAt 𝕜 d x → deriv (fun y => c y * d y) x = deriv c x * d x + c x * deriv d x- Defined in
- Mathlib.Analysis.Calculus.Deriv.Mul
- Cited by
- 15 results in Mathlib
- Foundations
- Depth 185 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- NormedAlgebrastatement and proof · cited by 1,165
- NormedRingstatement and proof · cited by 924
- derivstatement · cited by 676
- DifferentiableAtstatement and proof · cited by 617
- HasDerivAt.derivproof · cited by 147
- DifferentiableAt.hasDerivAtproof · cited by 114
- HasDerivAt.mulproof · cited by 20
Cited by15
Results whose statement or proof uses this declaration.
- ProbabilityTheory.integral_id_gaussianRealproof · cited by 6
- Real.deriv_mul_logproof · cited by 4
- logDeriv_mulproof · cited by 3
- ModularForm.logDeriv_one_sub_cexpproof · cited by 1
- DirichletCharacter.deriv_LFunctionTrivChar₁_apply_of_ne_oneproof · cited by 1
- Real.hasDerivAt_Gamma_one_halfproof · cited by 1
- Polynomial.deriv_gaussian_eq_hermite_mul_gaussianproof · cited by 1
- AkraBazziRecurrence.eventually_deriv_rpow_p_mul_one_add_smoothingFnproof · cited by 1
- AkraBazziRecurrence.eventually_deriv_rpow_p_mul_one_sub_smoothingFnproof · cited by 1
- AkraBazziRecurrence.rpow_p_mul_one_add_smoothingFn_geproof · cited by 1
- AkraBazziRecurrence.rpow_p_mul_one_sub_smoothingFn_leproof · cited by 1
- ProbabilityTheory.variance_fun_id_gaussianRealproof · cited by 1