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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,

02_differentiation-89.gif,   02_differentiation-90.gif

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,

02_differentiation-91.gif

Therefore

02_differentiation-92.gif

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.,

02_differentiation-93.gif

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.,

02_differentiation-94.gif       dc = 0.
Home Differentiation Derivatives of Rational Functions Theorem 1:

Last Update: 2010-11-25