case h.e'_7
f g : ℝ → ℝ
f' g' x y p : ℝ
s : Set ℝ
hf : HasDerivWithinAt f f' s x
hg : HasDerivWithinAt g g' s x
h : 0 < f x
⊢ f' * g x * f x ^ (g x - 1) + g' * f x ^ g x * Real.log (f x) =
    ↑((g x * f x ^ (g x - 1)) • ContinuousLinearMap.smulRight 1 f' +
          (f x ^ g x * Real.log (f x)) • ContinuousLinearMap.smulRight 1 g')
      1
f g : ℝ → ℝ
f' g' x y p : ℝ
s : Set ℝ
hf : HasDerivWithinAt f f' s x
hg : HasDerivWithinAt g g' s x
h : 0 < f x
⊢ HasDerivWithinAt (fun x => f x ^ g x) (f' * g x * f x ^ (g x - 1) + g' * f x ^ g x * Real.log (f x)) s x
convert (hf.hasFDerivWithinAt.rpow hg.hasFDerivWithinAt h).hasDerivWithinAt using 1
case h.e'_7
f g : ℝ → ℝ
f' g' x y p : ℝ
s : Set ℝ
hf : HasDerivWithinAt f f' s x
hg : HasDerivWithinAt g g' s x
h : 0 < f x
⊢ f' * g x * f x ^ (g x - 1) + g' * f x ^ g x * Real.log (f x) =
    g x * f x ^ (g x - 1) * (1 * f') + f x ^ g x * Real.log (f x) * (1 * g')
case h.e'_7
f g : ℝ → ℝ
f' g' x y p : ℝ
s : Set ℝ
hf : HasDerivWithinAt f f' s x
hg : HasDerivWithinAt g g' s x
h : 0 < f x
⊢ f' * g x * f x ^ (g x - 1) + g' * f x ^ g x * Real.log (f x) =
    ↑((g x * f x ^ (g x - 1)) • ContinuousLinearMap.smulRight 1 f' +
          (f x ^ g x * Real.log (f x)) • ContinuousLinearMap.smulRight 1 g')
      1
dsimp
no goals
case h.e'_7
f g : ℝ → ℝ
f' g' x y p : ℝ
s : Set ℝ
hf : HasDerivWithinAt f f' s x
hg : HasDerivWithinAt g g' s x
h : 0 < f x
⊢ f' * g x * f x ^ (g x - 1) + g' * f x ^ g x * Real.log (f x) =
    g x * f x ^ (g x - 1) * (1 * f') + f x ^ g x * Real.log (f x) * (1 * g')
ring