Differentiation from First Principles

The derivative measures how fast a function changes. Before any of the shortcut rules — the power rule, the product rule, the chain rule — there is a single definition that everything else is built on: the derivative from first principles. Understanding this definition is what separates students who can differentiate mechanically from students who understand what a derivative is.

Start with a function f(x) and two points on its graph: one at x and one a small distance h away, at x + h. The straight line joining these two points is called a secant, and its gradient is the change in the output divided by the change in the input: (f(x + h) − f(x)) / h. This fraction is the average rate of change of the function between the two points.

Now imagine sliding the second point closer and closer to the first, shrinking h toward zero. The secant line pivots until, in the limit, it becomes the tangent to the curve at x. The gradient of that tangent is the instantaneous rate of change — the derivative. Formally we write f′(x) = lim (h→0) (f(x + h) − f(x)) / h. The limit is essential: we cannot simply set h = 0, because that would give 0/0, which is undefined. Instead we simplify the fraction algebraically first, cancelling the h in the denominator, and only then let h approach zero.

Consider f(x) = x². Substituting into the definition gives ((x + h)² − x²) / h. Expanding the numerator: (x² + 2xh + h² − x²) / h, which simplifies to (2xh + h²) / h. Both terms in the numerator share a factor of h, so we can cancel it, leaving 2x + h. This is the crucial step — once the h in the denominator is gone, the expression is well behaved. Letting h → 0 now simply removes the last h, and we are left with f′(x) = 2x. That is exactly what the power rule would have told us, but now we know why it is true.

The same procedure works for any polynomial term. For f(x) = x³ you expand (x + h)³ = x³ + 3x²h + 3xh² + h³, subtract x³, divide by h to get 3x² + 3xh + h², and take the limit to reach 3x². A pattern is emerging: the derivative of xⁿ is n·xⁿ⁻¹. First principles is what proves the pattern rather than asking you to memorise it.

Two subtleties are worth stating clearly. First, the limit must exist and be the same whether h approaches zero from the positive or the negative side; if the two one-sided limits disagree, the function is not differentiable at that point — this is what happens at the sharp corner of y = |x| at x = 0. Second, differentiability is stronger than continuity: every differentiable function is continuous, but not every continuous function is differentiable. A curve can be unbroken yet still have a corner or a …