Derivative Rules from First Principles
The derivative of a function measures its instantaneous rate of change. Before applying any shortcut rule, it helps to see where those rules come from. This page derives the limit definition, the power rule, the sum rule, and the constant multiple rule — each from the same foundational limit.
The Limit Definition
The derivative of f at a point x is defined as:
Geometrically, the expression [f(x+h) − f(x)] / h is the slope of the secant line connecting (x, f(x)) and (x+h, f(x+h)). As h shrinks to zero, the secant line approaches the tangent line at x, and its slope approaches f′(x).
The derivative exists at x if and only if this limit is finite and the left-hand and right-hand limits agree.
The Power Rule: d/dx xⁿ = nxⁿ⁻¹
Let f(x) = xⁿ for a positive integer n. Expanding (x + h)ⁿ using the binomial theorem:
Substituting into the difference quotient:
Taking h → 0, all terms containing h vanish and we obtain:
The result also holds for any real exponent n (proven via logarithmic differentiation), not just positive integers.
The Sum Rule: (f + g)′ = f′ + g′
Let s(x) = f(x) + g(x). The difference quotient for s is:
Separating the two functions:
Since limits distribute over sums (when both limits exist), taking h → 0 gives:
The Constant Multiple Rule: (c·f)′ = c·f′
Let p(x) = c·f(x) for a constant c. The difference quotient is:
The constant c does not depend on h, so it factors outside the limit:
Worked Example
Find the derivative of f(x) = 3x⁴ + 5x² − 7.
- Apply the constant multiple rule: d/dx (3x⁴) = 3 · d/dx x⁴.
- Apply the power rule: d/dx x⁴ = 4x³, so d/dx (3x⁴) = 12x³.
- Apply the power rule again: d/dx x² = 2x, so d/dx (5x²) = 10x.
- The derivative of the constant 7 is 0.
- Apply the sum rule: f′(x) = 12x³ + 10x − 0 = 12x³ + 10x.
Frequently Asked Questions
What is the formal definition of a derivative?
The derivative of f at x is the limit f′(x) = lim_{h→0} [f(x+h) − f(x)] / h, provided the limit exists. This measures the instantaneous rate of change of f at the point x.
Why does d/dx xⁿ = nxⁿ⁻¹?
The power rule follows from the binomial expansion of (x+h)ⁿ: the first term is xⁿ, the second is nxⁿ⁻¹h, and all remaining terms contain h² or higher powers. Dividing by h and taking h→0 leaves exactly nxⁿ⁻¹.
What does the sum rule say, and why does it hold?
The sum rule states (f+g)′ = f′ + g′. It holds because the limit of a sum equals the sum of the limits: lim[f(x+h)+g(x+h) − f(x)−g(x)]/h splits into two independent difference quotients.
What is the constant multiple rule?
If c is a constant, then (cf)′ = c·f′. This follows because the constant c factors out of the limit, since lim c·[f(x+h)−f(x)]/h = c · lim [f(x+h)−f(x)]/h.
Adapted and rewritten from Silas Maths source notes; formulas reviewed for CalxSolver publication.