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Business Mathematics and Statistics · Ch 5 — Differential Calculus (Functions & Graphs, Limits & Derivatives, Differentiation Techniques)

Standard Derivatives Derived from First Principles

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Standard Derivatives Derived from First Principles

Working from the first-principles definition every single time would make calculus painfully slow, so a handful of standard results are derived once from first principles and then reused as ready-made formulae.

Derivative of a constant. If f(x)=cf(x)=c for a constant cc, then f(x+h)=cf(x+h)=c too, so

f′(x)=lim⁡h→0c−ch=lim⁡h→00=0f'(x) = \lim_{h\to0}\frac{c-c}{h} = \lim_{h\to0} 0 = 0

A constant function never changes, so its instantaneous rate of change is always zero — matching the algebra exactly.

Derivative of xnx^n (the power rule), for a positive integer nn. Using the binomial expansion of (x+h)n(x+h)^n:

f′(x)=lim⁡h→0(x+h)n−xnh=lim⁡h→0[xn+nxn−1h+(n2)xn−2h2+⋯+hn]−xnhf'(x) = \lim_{h\to0}\frac{(x+h)^n - x^n}{h} = \lim_{h\to0}\frac{\left[x^n + nx^{n-1}h + \binom{n}{2}x^{n-2}h^2 + \cdots + h^n\right] - x^n}{h}

Every remaining term in the bracket carries at least one factor of hh, so dividing by hh leaves

f′(x)=lim⁡h→0[nxn−1+(n2)xn−2h+⋯+hn−1]=nxn−1f'(x) = \lim_{h\to0}\left[nx^{n-1} + \binom{n}{2}x^{n-2}h + \cdots + h^{n-1}\right] = nx^{n-1}

since every term except the first contains a positive power of hh and vanishes as h→0h\to0. This gives the celebrated power rule:

ddx(xn)=n xn−1\frac{d}{dx}\left(x^n\right) = n\,x^{n-1} …

Definition 1Power Rule (a First-Principles Result)

ddx(xn)=n xn−1\dfrac{d}{dx}(x^n) = n\,x^{n-1}, proved for a positive integer nn by expanding (x+h)n(x+h)^n with the binomial theorem and letting h→0h\to0; the same formula holds …

Definition 2Derivative of a Constant

If f(x)=cf(x)=c (a constant) then f′(x)=0f'(x)=0 for every xx — a direct consequence of the first-principles limit, since $f(x+h …

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