Calculus Intuition — Derivatives and Integrals
The conceptual core of calculus — what derivatives and integrals mean, why they're inverses of each other, and where they appear.
What Calculus Is About
Calculus is the mathematics of continuous change. Two questions drive it:
- How fast is something changing right now? → Derivatives
- How much has accumulated over time? → Integrals
Everything else in calculus is an elaboration of these two ideas and their relationship.
The Derivative — Instantaneous Rate of Change
Consider a function f(x). The average rate of change between x and x+h is:
[f(x+h) − f(x)] / h
This is the slope of the secant line — the line through two points on the curve.
As h shrinks toward zero, the secant line approaches the tangent line at x. The limit of this ratio is the derivative:
f'(x) = lim[h→0] [f(x+h) − f(x)] / h
The derivative f’(x) is the instantaneous rate of change of f at x — the slope of the tangent to the curve at that point.
Notation: f’(x), df/dx, Df(x) all mean the same thing.
Derivatives of Common Functions
| Function | Derivative |
|---|---|
| c (constant) | 0 |
| xⁿ | nxⁿ⁻¹ |
| eˣ | eˣ |
| ln(x) | 1/x |
| sin(x) | cos(x) |
| cos(x) | −sin(x) |
| aˣ | aˣ ln(a) |
The power rule: d/dx(xⁿ) = nxⁿ⁻¹. Pull down the exponent, reduce it by 1.
d/dx(x³) = 3x²
d/dx(x¹⁰⁰) = 100x⁹⁹
d/dx(√x) = d/dx(x^½) = ½x^(−½) = 1/(2√x)
Rules of Differentiation
Sum rule: (f + g)’ = f’ + g’
Product rule: (fg)’ = f’g + fg’
Quotient rule: (f/g)’ = (f’g − fg’) / g²
Chain rule: if h(x) = f(g(x)), then h’(x) = f’(g(x)) · g’(x)
The chain rule is the most important. It handles compositions — the derivative of the outer function (evaluated at the inner) times the derivative of the inner.
d/dx[sin(x²)] = cos(x²) · 2x
d/dx[e^(3x)] = e^(3x) · 3
d/dx[(x² + 1)⁵] = 5(x² + 1)⁴ · 2x
What the Derivative Tells You
Sign of f’(x):
- f’(x) > 0: f is increasing at x
- f’(x) < 0: f is decreasing at x
- f’(x) = 0: f has a critical point (potential maximum, minimum, or inflection)
Finding maxima and minima: Set f’(x) = 0, solve. Check second derivative f”(x):
- f”(x) > 0: concave up → local minimum
- f”(x) < 0: concave down → local maximum
The second derivative measures curvature — how fast the slope is changing.
The Integral — Accumulated Change
The definite integral of f from a to b is the signed area under the curve:
∫ₐᵇ f(x) dx
Intuition: divide [a,b] into n thin strips of width Δx. Each strip has area ≈ f(x) · Δx. Sum all strips, let Δx → 0:
∫ₐᵇ f(x) dx = lim[n→∞] Σᵢ f(xᵢ) Δx
“Signed” because area below the x-axis counts negative.
The Fundamental Theorem of Calculus
Differentiation and integration are inverses. This is the central fact of calculus:
Part 1: if F(x) = ∫ₐˣ f(t) dt, then F’(x) = f(x). Differentiating an integral returns the original function.
Part 2: ∫ₐᵇ f(x) dx = F(b) − F(a), where F is any antiderivative of f (F’ = f).
∫₀³ x² dx = [x³/3]₀³ = 27/3 − 0 = 9
Why they’re inverses: differentiation asks “how fast is the area growing?” Integration asks “how much area has accumulated?” One undoes the other.
Antiderivatives and Integration Rules
The antiderivative F of f satisfies F’ = f. Also called the indefinite integral:
∫ f(x) dx = F(x) + C
The +C is necessary — any constant disappears on differentiation.
| Function | Antiderivative |
|---|---|
| xⁿ (n ≠ −1) | xⁿ⁺¹/(n+1) + C |
| 1/x | ln|x| + C |
| eˣ | eˣ + C |
| sin(x) | −cos(x) + C |
| cos(x) | sin(x) + C |
Substitution (chain rule in reverse): if the integrand contains a function and its derivative:
∫ 2x · cos(x²) dx let u = x², du = 2x dx
= ∫ cos(u) du = sin(u) + C = sin(x²) + C
Derivatives in Multiple Dimensions
For f(x, y, …) — a function of several variables:
Partial derivative ∂f/∂x: differentiate with respect to x, treating all other variables as constants.
f(x, y) = x²y + 3y
∂f/∂x = 2xy
∂f/∂y = x² + 3
Gradient: the vector of all partial derivatives. Points in the direction of steepest increase.
∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)
Gradient descent: move opposite to the gradient to find a minimum. Every neural network is trained by computing gradients and stepping downhill. Backpropagation is just the chain rule applied to a composition of many functions.
Where Calculus Appears
Physics: velocity = derivative of position; acceleration = derivative of velocity. Force = mass × acceleration. All of mechanics is derivatives.
Optimisation: find where derivative = 0 to locate maxima and minima. Economics, engineering, machine learning — all optimisation problems.
Probability: continuous probability distributions are defined by probability density functions; probabilities are integrals (areas under the curve).
Signal processing: Fourier transforms decompose signals into frequencies using integrals.
Machine learning: training a neural network means computing the gradient of a loss function with respect to millions of parameters, then stepping in the direction that reduces loss. Every weight update is calculus.
The Key Intuitions
Derivative = slope = rate of change. Whenever you see something changing, the derivative measures how fast.
Integral = area = accumulation. Whenever something builds up over time or space, the integral measures the total.
They undo each other. The derivative of an integral is the original function. The integral of a derivative is just the net change.
These three sentences contain most of what you need to reason about calculus without doing it formally.