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Home Differentiation Derivatives of Rational Functions Theorem 1: | |
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Theorem 1:
THEOREM 1 The derivative of a linear function is equal to the coefficient of x. That is, , PROOF Let y = bx + c, and let Δx ≠ 0 be infinitesimal. Then y + Δy = b(x + Δx) + c, Δy = (b(x + Δx) + c) - (bx + c) = b Δx, Therefore Multiplying through by dx, we obtain at once dy = b dx. If in Theorem 1 we put b = 1, c = 0, we see that the derivative of the identity function f(x) = x is f'(x) = 1; i.e., On the other hand, if we put b = 0 in Theorem 1 then the term bx + c is just the constant c, and we find that the derivative of the constant function f(x) = c is f'(x) = 0; i.e., dc = 0.
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Home Differentiation Derivatives of Rational Functions Theorem 1: |