How can I make the objToRotate speed to rotate depending on the target moving speed?

Question

The player has these components: Animator, Rigidbody, Capsule Collider, Third Person Character (Script), Third Person User Control (Script), and the MyLookAT script.

The problem is if I drag the cube target too fast in the editor changing its position it will take some time for the objToRotate to rotate facing the target again. I want somehow to make that the objToRotate will know to set his own speed and to rotate fast/slow enough to face the target when moving the target.

I want the objToRotate to face the target in real-time when the target is moving not with delay.

using UnityEngine;
using System;
using System.Collections;

public class RotatingTest : MonoBehaviour
{
    public Transform target;
    public Transform objToRotate;
    public float speed = 1.0f;

    void LateUpdate()
    {
        Vector3 targetDirection = target.position - objToRotate.position;
        float singleStep = speed * Time.deltaTime;

        Vector3 newDirection = Vector3.RotateTowards(objToRotate.forward, targetDirection, singleStep, 0.0f);

        Debug.DrawRay(objToRotate.position, newDirection, Color.red);

        objToRotate.rotation = Quaternion.LookRotation(newDirection);
    }
}

Answer 1

using UnityEngine;
using System;
using System.Collections;

[RequireComponent(typeof(Animator))]
public class MyLookAt : MonoBehaviour
{
    public Transform target;
    public Transform objToRotate;

    void LateUpdate()
    {
        Vector3 relativePos = target.position - objToRotate.position;
        Quaternion rotation = Quaternion.LookRotation(relativePos, Vector3.up);
        objToRotate.rotation = rotation;
    }
}