Can the gravity pendulum formula be derived using conservation of energy laws

I've look on the internet for this but all i could find are people deriving it using moment and force. I figured it would make sense logically to be able to derive it in this way and i have tried myself though my answer for gravitational acceleration ends up being a bit off.

Thanks to anyone who can help me :)

Yes, this is a fairly common way to derive it, in fact, because it becomes a technique that is useful to apply to other SHO's (simple harmonic oscillators).

The idea boils down to approximating the force in the neighborhood of an equilibrium point as being linear wrt to displacement from that point, and directed opposite to the displacement, so that the potential energy associated with that force is an upward-opening quadratic whose minimum is at the equilib. pt.

In the case of the pendulum, you assume a relatively small angular amplitude, so you can make the approximations:
sinθ ≈ θ
1 - cosθ ≈ ½θ²
with θ = s/L, where s = displacement from dead-hang, and L = length of pendulum

V(s) = MgL(1 - cosθ) ≈ ½MgLθ² = ½MgL s²/L² = Mg s²/2L

Then take the s=0 point as your reference potential, V(s=0) = 0, and at the ends of the swing,
|θ| = θ_max
|s| = s_max = A = (displacement) amplitude
v = 0; T(A) = ½Mv² = 0
V(A) ≈ Mg A²/2L

Now apply conservation of energy:
E = T + V = constant
≈ Mg A²/2L
≈½Mv² + Mg s²/2L
ds/dt = v = √[g(A² - s²)/L]

Try s = A sin(ωt+ϕ)
ds/dt = ω A cos(ωt+ϕ)
A² - s² = A² [1 - sin²(ωt+ϕ)] = A² cos²(ωt+ϕ)
So
ds/dt = ω A cos(ωt+ϕ) = √[g(A² - s²)/L] = √[g/L] A cos(ωt+ϕ)

ω = √[g/L];
and the amplitude, A, and phase, ϕ, are determined by initial conditions.
Period is T = 2π/ω = 2π√[L/g]

Good, I'm glad!
&Thanx!

The way I was taught, this is why the SHO pops up so much in physical systems -- all you need is a local minimum in the (smooth) potential function, and there will automatically be a SHO associated with that, arising from the quadratic approximation to the potential there.

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Why not. Conservation of energy relates the height to the speed of the pendulum. The height can be related to the angular deviation and thus to the position along the arc of movement (a circle) Thus, you know the velocity at every point along the arc without having to consider acceleration (or force). Integrating the velocity function will give position against time, and thus, you may determine the period.