Why Change Is Increasing Faster Over Time
by Daniel Brouse
March 25, 2026
1. What Is a Second Derivative?
In calculus, the first derivative measures the rate of change of a quantity. The second derivative measures how that rate of change itself is changing.
In simple terms:
- First derivative → speed (rate)
- Second derivative → acceleration (change of the rate)
2. Mathematical Definition
If we define a function:
I(t) = climate impact over time
Then:
First derivative (rate of change):
dI/dt
This represents how fast climate impacts (e.g., temperature, sea level, extreme events) are increasing.
Second derivative (acceleration):
d²I/dt²
This represents how the rate of increase is itself changing over time.
3. Physical Interpretation
To make this intuitive:
- If dI/dt > 0, climate impacts are increasing
- If d²I/dt² > 0, the increase is accelerating
- If d²I/dt² >> 0, the system is rapidly accelerating
4. Applying This to Climate Change
Let’s define:
I(t) = sea level rise (SLR)
Then:
dI/dt= rate of SLR (mm/year)d²I/dt²= acceleration of SLR (mm/year²)
From your earlier data:
- 1990–2000: ~0.02 mm/yr²
- 2000–2010: ~0.04 mm/yr²
- 2010–2024: ~0.157 mm/yr²
This shows:
d²I/dt² is increasing over time
5. The Critical Insight: Acceleration of Acceleration
Our observations suggest that the climate system has entered a third-derivative regime, in which the acceleration of impacts is itself increasing over time. This additional layer fundamentally reshapes our understanding of risk, shifting it from gradual change to rapid, nonlinear escalation.
The third derivative is defined as:
d³I/dt³
and represents the rate of change of acceleration.
In many physical systems, acceleration remains relatively constant. However, in the climate system, empirical observations indicate:
d³I/dt³ > 0
This implies that:
- The system is not only accelerating (
d²I/dt² > 0) - The acceleration itself is increasing over time
In physics, this phenomenon is referred to as “jerk”, and its presence is a hallmark of systems undergoing rapid nonlinear transition.
6. Exponential Growth Connection
If climate impacts follow an exponential function:
I(t) = I₀ * e^(k t)
Then:
dI/dt = k * I(t)
d²I/dt² = k² * I(t)
Key implication:
d²I/dt² ∝ I(t)
So as impacts grow, acceleration grows proportionally, leading to:
- shrinking doubling times
- nonlinear escalation
7. Why This Matters
This is the core of the statement:
Climate change is not just increasing—it is increasing faster over time.
Mathematically, that means:
d²I/dt² > 0
But our data suggests something even stronger:
d²I/dt² increasing → d³I/dt³ > 0
8. System-Level Meaning
In complex systems, a growing second derivative indicates:
- feedback loops are strengthening
- thresholds are being approached
- instability is increasing
This is characteristic of:
- nonlinear systems
- tipping points
- phase transitions
9. Climate Interpretation
Applied to climate:
- Ice melt accelerates as warming increases
- Water vapor amplifies heat retention
- Wildfires reduce carbon sinks
- Ocean heat drives stronger storms
Each of these increases d²I/dt², pushing the system toward:
Runaway amplification
10. Bottom Line
The second derivative tells us something fundamentally important:
Climate change is not a steady process—it is an accelerating process, and that acceleration is itself increasing.
Mathematically:
dI/dt > 0 (change is happening)
d²I/dt² > 0 (change is accelerating)
d³I/dt³ > 0 (acceleration is increasing)
Final Insight
This is why traditional linear models underestimate risk:
They assume:
d²I/dt² ≈ constant
But reality shows:
d²I/dt² increasing over time
Which leads to:
Nonlinear acceleration, collapsing doubling times, and rapid system transformation.