AnimationCurveExtensions.cs

/*

Copyright (c) 2018 Pete Michaud, github.com/PeteMichaud

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using System.Collections.Generic;
using System.Linq;
using UnityEngine;

public static class AnimationCurveExtensions
{

    /// <summary>
    /// Find first derivative of curve at point x
    /// </summary>
    /// <param name="curve"></param>
    /// <param name="x"></param>
    /// <returns>Slope of curve at point x as float</returns>
    public static float Differentiate(this AnimationCurve curve, float x) {
        return curve.Differentiate(x, curve.keys.First().time, curve.keys.Last().time);
    }

    const float Delta = .000001f;
    public static float Differentiate(this AnimationCurve curve, float x, float xMin, float xMax)
    {
        var x1 = Mathf.Max(xMin, x - Delta);
        var x2 = Mathf.Min(xMax, x + Delta);
        var y1 = curve.Evaluate(x1);
        var y2 = curve.Evaluate(x2);

        return (y2 - y1) / (x2 - x1);
    }


    static IEnumerable<float> GetPointSlopes(AnimationCurve curve, int resolution) {
        for (var i = 0; i < resolution; i++) {
            yield return curve.Differentiate((float)i / resolution);
        }
    }

    public static AnimationCurve Derivative(this AnimationCurve curve, int resolution = 100, float smoothing = .05f) {
        var slopes = GetPointSlopes(curve, resolution).ToArray();

        var curvePoints = slopes
            .Select((s, i) => new Vector2((float) i / resolution, s))
            .ToList();

        var simplifiedCurvePoints = new List<Vector2>();

        if (smoothing > 0) {
            LineUtility.Simplify(curvePoints, smoothing, simplifiedCurvePoints);
        } else {
            simplifiedCurvePoints = curvePoints;
        }

        var derivative = new AnimationCurve(
            simplifiedCurvePoints.Select(v => new Keyframe(v.x, v.y)).ToArray());

        for (int i = 0, len = derivative.keys.Length; i < len; i++) {
            derivative.SmoothTangents(i, 0);
        }

        return derivative;
    }

    /// <summary>
    /// Find integral of curve between xStart and xEnd using the trapezoidal rule
    /// </summary>
    /// <param name="curve"></param>
    /// <param name="xStart"></param>
    /// <param name="xEnd"></param>
    /// <param name="sliceCount">Resolution of calculation. Increase for better
    /// precision, at cost of computation</param>
    /// <returns>Area under the curve between xStart and xEnd as float</returns>
    public static float Integrate(this AnimationCurve curve, float xStart = 0f, float xEnd = 1f, int sliceCount = 100)
    {
        var sliceWidth = (xEnd - xStart) / sliceCount;
        var accumulatedTotal = (curve.Evaluate(xStart) + curve.Evaluate(xEnd)) / 2;

        for (var i = 1; i < sliceCount; i++) {
            accumulatedTotal += curve.Evaluate(xStart + i * sliceWidth);
        }

        return sliceWidth * accumulatedTotal;
    }

}

CurveTest.cs

using UnityEngine;

[ExecuteInEditMode]
public class CurveTest : MonoBehaviour
{
    //This is really non-performant code, don't use it like this.
    //It's just a quick demo for you to drop into a scene and get started
    
    public AnimationCurve SleepCurve = AnimationCurve.Linear(0f, 0f, 1f, 1f);
    public AnimationCurve SleepDerivative;
    public float SleepCurveIntegral;

    void Update() {
            SleepDerivative = SleepCurve.Derivative();
            SleepCurveIntegral = SleepCurve.Integrate();
    }

}

When testing this code I was getting some weird results when differentiating certain curves. Lots of noise in the resulting derivative. Increasing the delta to .001 stabilized the results significantly, so perhaps it's some kind of precision error with AnimationCurve.Evaluate.

Also taking the derivative of sine gives you 6*cosine, and the derivative of cosine results in 6*sine. So something's definitely off with this approach, and I'm not good enough at math to figure it out lol.