Derivative Definition

The limit definition

The derivative of a function f at a point x is defined as:

f'(x) = lim_{h→0}  (f(x + h) − f(x)) / h

If this limit exists, f is differentiable at x and f'(x) is its derivative. The function f' (taking x to f'(x)) is the derivative function.

Geometrically — (f(x+h) − f(x))/h is the slope of the line through (x, f(x)) and (x+h, f(x+h)) — the secant line. As h shrinks, this secant rotates toward the tangent line at x. The limit IS the slope of the tangent.

Computing a derivative from first principles

Let's compute d/dx(x²):

f(x) = x²
f(x+h) − f(x) = (x+h)² − x² = x² + 2xh + h² − x² = 2xh + h²

(f(x+h) − f(x))/h = (2xh + h²)/h = 2x + h     (assuming h ≠ 0)

f'(x) = lim_{h→0} (2x + h) = 2x

So the derivative of x² is 2x. The limit can be evaluated by direct substitution at the end because 2x + h has no singularity at h = 0 — but we couldn't substitute h = 0 in the original (f(x+h) − f(x))/h, which would give 0/0.

Standard derivatives — proven once, applied forever

Function f(x)	Derivative f'(x)	Notes
c (constant)	0	Constants don't change
x^n	n · x^(n−1)	The power rule
e^x	e^x	The unique self-deriving function
ln(x)	1/x	For x > 0
sin(x)	cos(x)	Radians required
cos(x)	−sin(x)	Radians required
a^x	a^x · ln(a)	Reduces to e^x when a = e
log_a(x)	1/(x ln a)	Reduces to 1/x when a = e

Each of these can be derived from the limit definition. Once derived, they're applied mechanically — never re-derived.

The four arithmetic rules

Rule	Formula
Sum	(f + g)' = f' + g'
Constant multiple	(c·f)' = c·f'
Product	(f·g)' = f'·g + f·g'
Quotient	(f/g)' = (f'·g − f·g') / g²
Chain rule	(f(g(x)))' = f'(g(x)) · g'(x)

Combined with the standard derivatives, these rules cover almost every function you'll meet. The derivation of each (from the limit definition) is a one-page exercise; once proven, you apply mechanically.

When derivatives don't exist

A function is non-differentiable at a point in three classic ways:

• Sharp corners. f(x) = |x| has slope +1 from the right, −1 from the left at x = 0. The two-sided limit doesn't agree; not differentiable at 0.
• Vertical tangent. f(x) = x^(1/3) has infinite slope at x = 0. Limit is +∞, not finite; not differentiable.
• Discontinuity. Any jump or removable discontinuity destroys differentiability automatically (continuity is required).
• Pathological cases. Functions like Weierstrass's W(x) = Σ a^n cos(b^n πx) are continuous everywhere but differentiable nowhere — they're "fractal" with self-similar wiggling at all scales.

JavaScript: numerical differentiation

// Central difference — better than forward difference
function derivative(f, x, h = 1e-6) {
  return (f(x + h) - f(x - h)) / (2 * h);
}

derivative(x => x*x, 3);          // ≈ 6 (analytical: 2·3 = 6)
derivative(Math.sin, Math.PI/4);  // ≈ 0.707 (analytical: cos(π/4) = √2/2)
derivative(Math.exp, 1);          // ≈ 2.718 (analytical: e^1 = e)

// Bigger h: less precision but faster convergence to the analytical
// Smaller h: more precision but eventual catastrophic cancellation around h ≈ 1e-8

// For symbolic derivatives, you'd need a CAS like math.js or sympy.
// For real production code, autodiff libraries (TensorFlow, JAX, PyTorch)
// compute exact derivatives by tracing operations.

Why the derivative matters

• Velocity is the derivative of position. v(t) = ds/dt. Acceleration is the derivative of velocity. Newton's second law uses derivatives — F = m · d²s/dt².
• Optimization. Maxima and minima of differentiable functions occur where f'(x) = 0 (critical points). Solve for them, classify, and you've found extrema.
• Marginal economics. Marginal cost is the derivative of total cost. Marginal revenue is the derivative of total revenue. Microeconomics uses derivatives constantly.
• Approximation. Linearizing — f(x) ≈ f(a) + f'(a)(x − a) — is the basis of Newton's method, Taylor series, and most numerical algorithms.
• Differential equations. Models in physics, biology, engineering all express relationships between quantities and their rates of change. The derivative is the language.
• Machine learning — gradient descent. Compute ∂Loss/∂weights for every weight in a neural network; update weights against the gradient. Modern AI runs on derivatives.

Newton vs Leibniz

Both invented calculus independently in the 1660s-1670s. Newton's approach was geometric and physical — instantaneous velocity, motion under gravity. Leibniz's was symbolic and algebraic — the dy/dx and ∫ notation we still use today.

The bitter priority dispute (Royal Society found Leibniz had plagiarized; modern historians disagree) split mathematical traditions for a century. English mathematicians used Newton's awkward "fluxion" notation; Continental mathematicians used Leibniz's elegant differentials and made faster progress in the 18th century. By 1820, England had switched too. The notational war was decided; we use Leibniz's everywhere.

Common mistakes

• Trying to evaluate the difference quotient at h = 0. That gives 0/0, undefined. The limit operation is what extracts the meaningful value as h approaches 0.
• Treating dy/dx as a fraction. Sometimes it works (chain rule looks like fractions); sometimes it fails (you can't multiply by dx as if dividing). Use the rules; treat dy/dx as one symbol unless you've studied differential forms.
• Forgetting that radians are required. d/dx(sin x) = cos x ONLY when x is in radians. In degrees, the derivative gets a factor of π/180. This is why pure mathematics always uses radians.
• Not checking continuity before differentiating. A function not continuous at a point isn't differentiable there. Sharp jumps, removable discontinuities, and asymptotes all break differentiability.
• Numerical differentiation with too-small h. Floating-point round-off dominates when h is below ~10^−8. The "optimal" h depends on f and machine precision; central difference (f(x+h) − f(x−h))/(2h) is more stable than forward difference.
• Confusing derivative and antiderivative. The derivative is one direction; integration is the inverse. d/dx(x²) = 2x. ∫2x dx = x² + C. The constant C disappears under differentiation, recovering the loss of information.