Taylor series operator commutes with differentiation: Difference between revisions
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==Statement== | ==Statement== | ||
Suppose <math>f</math> if a function defined on a subset of the reals that is [[infinitely differentiable function|infinitely differentiable]] at a point <math>x_0/math> in its [[domain]]. Then, the [[derivative]] <math>f'</math> is also defined and infinitely differentiable at <math>x_0</math>, and the [[fact about::Taylor series]] for <math>f'</math> is the [[fact about::derivative of power series|derivative]] (in the sense of derivative of power series) of the Taylor series for <math>f</math>. | Suppose <math>f</math> if a function defined on a subset of the reals that is [[infinitely differentiable function|infinitely differentiable]] at a point <math>x_0</math> in its [[domain]]. Then, the [[derivative]] <math>f'</math> is also defined and infinitely differentiable at <math>x_0</math>, and the [[fact about::Taylor series]] for <math>f'</math> is the [[fact about::derivative of power series|derivative]] (in the sense of derivative of power series) of the Taylor series for <math>f</math>. | ||
==Related facts== | ==Related facts== | ||
Revision as of 11:43, 7 September 2011
Statement
Suppose if a function defined on a subset of the reals that is infinitely differentiable at a point in its domain. Then, the derivative is also defined and infinitely differentiable at , and the Taylor series for is the derivative (in the sense of derivative of power series) of the Taylor series for .