Definition D1
f is differentiable at x means that the limit lim (h→0) [f(x+h) − f(x)]/h exists, and its value is written f′(x).
Proof.
(1)a − b means a + (−b), and F/h means F · h⁻¹. The difference quotient is defined only for h ≠ 0.convention
(2)If the limit exists, its value is unique. Therefore the notation f′(x) is justified.A7
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Definition D2
For a natural number n, set x¹ = x and x^(k+1) = xᵏ · x.
Proof.
Therefore, if k ≥ 2, then hᵏ = h² · h^(k−2).convention, A12
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Lemma L1
If n is a natural number and h ≠ 0, there is a polynomial R such that [(x+h)ⁿ − xⁿ]/h = nxⁿ⁻¹ + h · R(x,h).
Proof.
(1)By the binomial theorem, (x+h)ⁿ = Σ (k=0…n) nCk xⁿ⁻ᵏ hᵏA1, D2
(2)The k = 0 term is nC0 xⁿ = xⁿ, and the k = 1 term is nC1 xⁿ⁻¹h = n xⁿ⁻¹ h.(1), A2, A15
(3)For k ≥ 2, hᵏ = h² h^(k−2), so every such term has h² as a factor. Write the sum of those terms as h² R(x,h).(1), D2, A11, A12, A13
(4)Therefore (x+h)ⁿ = xⁿ + n xⁿ⁻¹ h + h² R(x,h)(1), (2), (3), A17, A18
(5)Subtracting xⁿ from both sides gives (x+h)ⁿ − xⁿ = n xⁿ⁻¹ h + h² R(x,h)(4), E, A17, A18, A19, A20
(6)h ≠ 0, so multiplying both sides by h⁻¹ gives [(x+h)ⁿ − xⁿ]/h = n xⁿ⁻¹ + h R(x,h)(5), E, A11, A12, A13, A14, A15
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Theorem T1
If n is a natural number, f(x) = xⁿ is differentiable at every x and f′(x) = n xⁿ⁻¹.
Proof.
(1)For h ≠ 0, the difference quotient of f is n xⁿ⁻¹ + h R(x,h).L1, D1
(2)lim (h→0) h = 0A5
(3)R(x,h) is a polynomial in h, so each term is a constant times a power of h, and its limit exists.A3, A4, A5, A6, D2
(4)Therefore lim (h→0) h R(x,h) = 0 · lim R(x,h) = 0(2), (3), A4, A16
(5)lim (h→0) n xⁿ⁻¹ = n xⁿ⁻¹A6
(6)Adding the two limits gives lim (h→0) [n xⁿ⁻¹ + h R(x,h)] = n xⁿ⁻¹(4), (5), A3, A20
(7)Therefore f is differentiable at x and f′(x) = n xⁿ⁻¹.(1), (6), A7, D1
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Corollary C
If m is a natural number and x ≠ 0, then (x⁻ᵐ)′ = −m x⁻ᵐ⁻¹.
Proof.
(1)x⁻ᵐ means (xᵐ)⁻¹.A9
(2)Let g(x) = xᵐ; then g′(x) = m xᵐ⁻¹.T1
(3)By the quotient rule, (x⁻ᵐ)′ = (1/g)′ = −g′/g² = −m xᵐ⁻¹ · ((xᵐ)²)⁻¹(1), (2), A8
(4)(xᵐ)² = xᵐ xᵐ = x²ᵐ, and xᵐ⁻¹ · (x²ᵐ)⁻¹ = xᵐ⁻¹ x⁻²ᵐ = x⁻ᵐ⁻¹.A10, A9, D2
(5)Therefore (x⁻ᵐ)′ = −m x⁻ᵐ⁻¹.(3), (4), A12, A13
(6)The meaning of x⁻ᵐ is a convention chosen so that the law of exponents is preserved. Under that convention, the form of the derivative also matches that of natural-number exponents.(5), A9, A10
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