The Fundamental Theorem of Calculus
“Differentiation and integration are a pair that undo each other.”
The formula
d/dx ∫ₐˣ f(t) dt = f(x)How to read it: differentiate the area-so-far function and you return to the original function — integration and differentiation are inverse operations
- ∫ₐˣ f(t) dt
- — the area piled up from a to x (the accumulation function)
- d/dx
- — differentiate that area function
- f(x)
- — and you land back on the original function
The hook
Two ideas that grew up apart — cutting (differentiation) and stacking (integration) — turn out to be a pair that undoes each other. That's the great discovery of calculus.
In plain words
Differentiate the function that has been accumulating area from a to x, and the original function f(x) pops right back out. Differentiation and integration are inverses.
The intuition
Pour water into a bucket. Differentiate the amount held so far (the integral, the accumulation) with respect to time, and you get the pouring speed right now (the original function). The rate of change of the total piled up = the current value.
How it's built
It comes in two parts — ① differentiating the area-accumulation function returns f, and ② therefore a definite integral can be computed as F(b)−F(a). The antiderivative that differentiation hands you is what cracks the integral.
Example
F(x)=∫₀ˣ t dt is the area of a triangle with base x and height x, so x²/2. Differentiate it: d/dx(x²/2)=x, landing exactly back on the original f(t)=t.
Common mistake
People often think differentiation and integration are 'two unrelated procedures'. In fact they cancel each other like addition and subtraction — inverse operations.
Where it's used
Thanks to this theorem you don't add up areas forever; find a single antiderivative and the definite integral falls out — the beating heart of practical calculus.
Where it came from
Isolated pieces were known to predecessors like Barrow (Newton's teacher), but tying differentiation and integration into one theorem was the work of Newton and Leibniz.
Prerequisites
Quick check
Which two operations does the Fundamental Theorem tie together?
- addition and multiplication
- differentiation and integration✓
- limits and continuity
- product and quotient
Practice
For g(x)=∫₀ˣ t² dt, use the theorem to find g'(3).
Answer: 9
- By the theorem: g'(x)=x²
- Plug in x=3 → 9
Takeaway: Differentiate the accumulation function and the inside function pops right out.
Use the theorem to evaluate ∫₀⁴ 2t dt.
Answer: 16
- Antiderivative: t²
- [t²]₀⁴ = 16 − 0 = 16
Takeaway: Find the antiderivative and the area comes out in one subtraction.
What is d/dx ∫₁ˣ (t³+1) dt? Write one line on why.
Answer: undefined
- By the theorem, = x³+1
- Because differentiating the area-so-far gives the value being poured in right now
Takeaway: The lower limit (1) doesn't affect the derivative — only the upper limit x matters.