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Theorem 2: Sum Rule

THEOREM 2 (Sum Rule)

Suppose u and v depend on the independent variable x. Then for any value of x where du/dx and dv/dx exist,

02_differentiation-95.gif ,      d(u + v) - du + dv.

In other words, the derivative of the sum is the sum of the derivatives.

PROOF Let y = u + v, and let Δx ≠ 0 be infinitesimal. Then

02_differentiation-96.gif

Taking standard parts,

02_differentiation-97.gif

Thus

02_differentiation-98.gif

By using the Sum Rule n - 1 times, we see that

02_differentiation-99.gif


Last Update: 2010-11-25