Theorem: SciLean.Scalar.sqrt.arg_x.fwdDeriv_rule (SciLean/Core/Functions/Pow.lean)

State before:
R : Type u_1
inst✝² : RealScalar R
W : Type u_2
inst✝¹ : Vec R W
U : Type ?u.13231
inst✝ : SemiInnerProductSpace R U
x : W → R
hx : CDifferentiable R x
hw : ∀ (w : W), 0 < x w
⊢ (fwdDeriv R fun w => sqrt (x w)) = fun w dw =>
    let xdx := fwdDeriv R x w dw;
    let x' := sqrt xdx.1;
    (x', xdx.2 / (2 * x'))
Tactic:
unfold fwdDeriv
State after:
R : Type u_1
inst✝² : RealScalar R
W : Type u_2
inst✝¹ : Vec R W
U : Type ?u.13231
inst✝ : SemiInnerProductSpace R U
x : W → R
hx : CDifferentiable R x
hw : ∀ (w : W), 0 < x w
⊢ (fun x_1 dx => ((fun w => sqrt (x w)) x_1, cderiv R (fun w => sqrt (x w)) x_1 dx)) = fun w dw =>
    let xdx := (x w, cderiv R x w dw);
    let x' := sqrt xdx.1;
    (x', xdx.2 / (2 * x'))

State before:
R : Type u_1
inst✝² : RealScalar R
W : Type u_2
inst✝¹ : Vec R W
U : Type ?u.13231
inst✝ : SemiInnerProductSpace R U
x : W → R
hx : CDifferentiable R x
hw : ∀ (w : W), 0 < x w
⊢ (fun x_1 dx => ((fun w => sqrt (x w)) x_1, cderiv R (fun w => sqrt (x w)) x_1 dx)) = fun w dw =>
    let xdx := (x w, cderiv R x w dw);
    let x' := sqrt xdx.1;
    (x', xdx.2 / (2 * x'))
Tactic:
fun_trans (disch:=assumption) [fwdDeriv]
State after:
no goals

State before:
case funProp.discharger
R : Type u_1
inst✝² : RealScalar R
W : Type u_2
inst✝¹ : Vec R W
U : Type ?u.13231
inst✝ : SemiInnerProductSpace R U
x✝ : W → R
hx : CDifferentiable R x✝
hw : ∀ (w : W), 0 < x✝ w
x dx : W
⊢ ∀ (w : W), 0 < x✝ w
Tactic:
assumption
State after:
no goals