File: Mathlib/Analysis/Calculus/Deriv/Polynomial.lean

full_name: Polynomial.hasStrictDerivAt
code: /-- The derivative (in the analysis sense) of a polynomial `p` is given by `p.derivative`. -/
protected theorem hasStrictDerivAt (x : 𝕜) :
    HasStrictDerivAt (fun x => p.eval x) (p.derivative.eval x) x 
goal: HasStrictDerivAt (fun x => Polynomial.eval x p) (Polynomial.eval x (Polynomial.derivative p)) x

full_name: Polynomial.hasStrictDerivAt_aeval
code: protected theorem hasStrictDerivAt_aeval (x : 𝕜) :
    HasStrictDerivAt (fun x => aeval x q) (aeval x (derivative q)) x 
goal: HasStrictDerivAt (fun x => (Polynomial.aeval x) q) ((Polynomial.aeval x) (Polynomial.derivative q)) x

full_name: Polynomial.hasDerivAt
code: /-- The derivative (in the analysis sense) of a polynomial `p` is given by `p.derivative`. -/
protected theorem hasDerivAt (x : 𝕜) : HasDerivAt (fun x => p.eval x) (p.derivative.eval x) x 
goal: HasDerivAt (fun x => Polynomial.eval x p) (Polynomial.eval x (Polynomial.derivative p)) x

full_name: Polynomial.hasDerivAt_aeval
code: protected theorem hasDerivAt_aeval (x : 𝕜) :
    HasDerivAt (fun x => aeval x q) (aeval x (derivative q)) x 
goal: HasDerivAt (fun x => (Polynomial.aeval x) q) ((Polynomial.aeval x) (Polynomial.derivative q)) x

full_name: Polynomial.hasDerivWithinAt
code: protected theorem hasDerivWithinAt (x : 𝕜) (s : Set 𝕜) :
    HasDerivWithinAt (fun x => p.eval x) (p.derivative.eval x) s x 
goal: HasDerivWithinAt (fun x => Polynomial.eval x p) (Polynomial.eval x (Polynomial.derivative p)) s x

full_name: Polynomial.hasDerivWithinAt_aeval
code: protected theorem hasDerivWithinAt_aeval (x : 𝕜) (s : Set 𝕜) :
    HasDerivWithinAt (fun x => aeval x q) (aeval x (derivative q)) s x 
goal: HasDerivWithinAt (fun x => (Polynomial.aeval x) q) ((Polynomial.aeval x) (Polynomial.derivative q)) s x

full_name: Polynomial.differentiableAt
code: protected theorem differentiableAt : DifferentiableAt 𝕜 (fun x => p.eval x) x 
goal: DifferentiableAt 𝕜 (fun x => Polynomial.eval x p) x

full_name: Polynomial.differentiableAt_aeval
code: protected theorem differentiableAt_aeval : DifferentiableAt 𝕜 (fun x => aeval x q) x 
goal: DifferentiableAt 𝕜 (fun x => (Polynomial.aeval x) q) x

full_name: Polynomial.differentiableWithinAt
code: protected theorem differentiableWithinAt : DifferentiableWithinAt 𝕜 (fun x => p.eval x) s x 
goal: DifferentiableWithinAt 𝕜 (fun x => Polynomial.eval x p) s x

full_name: Polynomial.differentiableWithinAt_aeval
code: protected theorem differentiableWithinAt_aeval :
    DifferentiableWithinAt 𝕜 (fun x => aeval x q) s x 
goal: DifferentiableWithinAt 𝕜 (fun x => (Polynomial.aeval x) q) s x

full_name: Polynomial.differentiable
code: protected theorem differentiable : Differentiable 𝕜 fun x => p.eval x 
goal: Differentiable 𝕜 fun x => Polynomial.eval x p

full_name: Polynomial.differentiable_aeval
code: protected theorem differentiable_aeval : Differentiable 𝕜 fun x : 𝕜 => aeval x q 
goal: Differentiable 𝕜 fun x => (Polynomial.aeval x) q

full_name: Polynomial.differentiableOn
code: protected theorem differentiableOn : DifferentiableOn 𝕜 (fun x => p.eval x) s 
goal: DifferentiableOn 𝕜 (fun x => Polynomial.eval x p) s

full_name: Polynomial.differentiableOn_aeval
code: protected theorem differentiableOn_aeval : DifferentiableOn 𝕜 (fun x => aeval x q) s 
goal: DifferentiableOn 𝕜 (fun x => (Polynomial.aeval x) q) s

full_name: Polynomial.deriv
code: @[simp]
protected theorem deriv : deriv (fun x => p.eval x) x = p.derivative.eval x 
goal: deriv (fun x => Polynomial.eval x p) x = Polynomial.eval x (Polynomial.derivative p)

full_name: Polynomial.deriv_aeval
code: @[simp]
protected theorem deriv_aeval : deriv (fun x => aeval x q) x = aeval x (derivative q) 
goal: deriv (fun x => (Polynomial.aeval x) q) x = (Polynomial.aeval x) (Polynomial.derivative q)

full_name: Polynomial.derivWithin
code: protected theorem derivWithin (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin (fun x => p.eval x) s x = p.derivative.eval x 
goal: derivWithin (fun x => Polynomial.eval x p) s x = Polynomial.eval x (Polynomial.derivative p)

full_name: Polynomial.derivWithin_aeval
code: protected theorem derivWithin_aeval (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin (fun x => aeval x q) s x = aeval x (derivative q) 
goal: derivWithin (fun x => (Polynomial.aeval x) q) s x = (Polynomial.aeval x) (Polynomial.derivative q)

full_name: Polynomial.hasFDerivAt
code: protected theorem hasFDerivAt (x : 𝕜) :
    HasFDerivAt (fun x => p.eval x) (smulRight (1 : 𝕜 →L[𝕜] 𝕜) (p.derivative.eval x)) x 
goal: HasFDerivAt (fun x => Polynomial.eval x p)
  (ContinuousLinearMap.smulRight 1 (Polynomial.eval x (Polynomial.derivative p))) x

full_name: Polynomial.hasFDerivAt_aeval
code: protected theorem hasFDerivAt_aeval (x : 𝕜) :
    HasFDerivAt (fun x => aeval x q) (smulRight (1 : 𝕜 →L[𝕜] 𝕜) (aeval x (derivative q))) x 
goal: HasFDerivAt (fun x => (Polynomial.aeval x) q)
  (ContinuousLinearMap.smulRight 1 ((Polynomial.aeval x) (Polynomial.derivative q))) x

full_name: Polynomial.hasFDerivWithinAt
code: protected theorem hasFDerivWithinAt (x : 𝕜) :
    HasFDerivWithinAt (fun x => p.eval x) (smulRight (1 : 𝕜 →L[𝕜] 𝕜) (p.derivative.eval x)) s x 
goal: HasFDerivWithinAt (fun x => Polynomial.eval x p)
  (ContinuousLinearMap.smulRight 1 (Polynomial.eval x (Polynomial.derivative p))) s x

full_name: Polynomial.hasFDerivWithinAt_aeval
code: protected theorem hasFDerivWithinAt_aeval (x : 𝕜) :
    HasFDerivWithinAt (fun x => aeval x q) (smulRight (1 : 𝕜 →L[𝕜] 𝕜)
      (aeval x (derivative q))) s x 
goal: HasFDerivWithinAt (fun x => (Polynomial.aeval x) q)
  (ContinuousLinearMap.smulRight 1 ((Polynomial.aeval x) (Polynomial.derivative q))) s x

full_name: Polynomial.fderiv
code: @[simp]
protected theorem fderiv :
    fderiv 𝕜 (fun x => p.eval x) x = smulRight (1 : 𝕜 →L[𝕜] 𝕜) (p.derivative.eval x) 
goal: fderiv 𝕜 (fun x => Polynomial.eval x p) x =
  ContinuousLinearMap.smulRight 1 (Polynomial.eval x (Polynomial.derivative p))

full_name: Polynomial.fderiv_aeval
code: @[simp]
protected theorem fderiv_aeval :
    fderiv 𝕜 (fun x => aeval x q) x = smulRight (1 : 𝕜 →L[𝕜] 𝕜) (aeval x (derivative q)) 
goal: fderiv 𝕜 (fun x => (Polynomial.aeval x) q) x =
  ContinuousLinearMap.smulRight 1 ((Polynomial.aeval x) (Polynomial.derivative q))

full_name: Polynomial.fderivWithin
code: protected theorem fderivWithin (hxs : UniqueDiffWithinAt 𝕜 s x) :
    fderivWithin 𝕜 (fun x => p.eval x) s x = smulRight (1 : 𝕜 →L[𝕜] 𝕜) (p.derivative.eval x) 
goal: fderivWithin 𝕜 (fun x => Polynomial.eval x p) s x =
  ContinuousLinearMap.smulRight 1 (Polynomial.eval x (Polynomial.derivative p))

full_name: Polynomial.fderivWithin_aeval
code: protected theorem fderivWithin_aeval (hxs : UniqueDiffWithinAt 𝕜 s x) :
    fderivWithin 𝕜 (fun x => aeval x q) s x = smulRight (1 : 𝕜 →L[𝕜] 𝕜) (aeval x (derivative q)) 
goal: fderivWithin 𝕜 (fun x => (Polynomial.aeval x) q) s x =
  ContinuousLinearMap.smulRight 1 ((Polynomial.aeval x) (Polynomial.derivative q))