When Calculus was first discovered, there was a war over what to call it.
Isaac Newton, the Englishman, viewed motion as a continuous "Flow." He used a simple dot or a "prime" mark to show the speed. He wrote **$y'$** (y-prime). This was fast, clean, and efficient.
Gottfried Leibniz, the German, viewed motion as a "Ratio" of tiny changes. He wrote **$\frac{dy}{dx}$**. This looked clunky, like a fraction. But it was powerful. It reminded everyone that speed comes from *Change* divided by *Change*.
In the end, the world decided to keep Both. We need Newton's speed for writing equations, but we need Leibniz's precision for understanding the mechanism.
In the Kingdom, we also have two names for the One who changes us. He is the **Son** (Relation) and the **Word** (Definition). $f'(x)$ is the Son; $\frac{dy}{dx}$ is the Word.
It is a Function.
Just like $f(x)$ is a machine that spits out "Height," $f'(x)$ is a machine that spits out "Slope."
If you plug a number into $f(x)$, you ask: "Where am I?"
If you plug a number into $f'(x)$, you ask: "How fast am I changing?"
This is the Dual Awareness of the believer. We must know our Position in Christ ($f$) and our Rate of Growth in the Spirit ($f'$).
Can you find the slope of a sharp corner?
Imagine a V-shape. On the left side, the slope is -1 (Down). On the right side, the slope is +1 (Up).
But at the exact tip of the V... is it -1 or +1?
It cannot be both. Therefore, at a Sharp Corner, the derivative does not exist.
We say the function is Not Differentiable.
This is a powerful lesson. God's grace flows best through Smooth curves. If our lives are full of sharp, jerky reactions—anger one moment, praise the next—it is hard for the "Derivative of Grace" to be defined in us. We need the "oil of the Spirit" to round off our sharp edges so we can carry the motion of Heaven.
The sign (+ or -) of the derivative tells you the story of the curve.
When you feel like you are standing still (Stationary), do not despair! A slope of zero often means you have reached a Summit. It is the moment before a new direction begins.
If $f(x) = 5x + 3$ (a line), the slope is always 5.
So $f'(x) = 5$.
Notice that the derivative is a Constant. A linear life has a constant zeal. It does not waver. This is the "Faithfulness" of the saints.
You can keep going!
The derivative of Velocity is Acceleration ($f''$).
The derivative of Acceleration is Jerk ($f'''$).
Acceleration tells us if the "Change is Changing."
Are you growing *faster* than you were yesterday? That is spiritual acceleration.
"I vow to be a differentiable vessel. I ask the Holy Spirit to smooth the sharp corners of my character—my pride, my anger, my double-mindedness—so that His power may flow through me without interruption. I will monitor my 'Signs' ($f'$), rejoicing in growth, accepting pruning, and waiting patiently in the zero-slope moments of rest."
The "Notation Wars" between Leibniz and Newton effectively divided Europe for a century. The British stuck to Newton's dots and fell behind in mathematics, while the Continent used Leibniz's differentials and surged ahead. This is a lesson in "Form and Function." Newton's form was simpler, but Leibniz's form carried more meaning. It showed the ratio. In our spiritual lives, we often prefer the "Simple Dot"—just tell me what to do! But God often gives us the "Differential Ratio"—He wants us to understand why and how the change is happening. He wants us to see the components ($\Delta y$ and $\Delta x$) so we can apply wisdom, not just rote obedience.
The concept of "Non-Differentiability" at a cusp (sharp point) is physically relevant. If a roller coaster track has a cusp, the G-forces become infinite at that point. The structure fails. The math "breaks" because reality "breaks." This confirms that mathematics is the language of physical reality. When the math says "Undefined," nature says "Broken." This is why God shapes us. He is removing the cusps so that when He pours His infinite power through us, we don't break under the stress. Sanctification is the process of making us differentiable.
Finally, the hierarchy of derivatives ($f, f', f''$) suggests a "Deep Structure" to time. We usually only see $f$ (where things are). Sometimes we feel $f'$ (how fast things are moving). But $f''$ (acceleration) is often invisible until it's too late. It is the force that pushes you back in your seat. A leader must learn to sense the $f''$ of a culture or a family. Is the tension accelerating? Is the love accelerating? Detecting the rate of change of the rate of change is the ultimate prophetic sensitivity.