Differentials

    Let where is a differentiable function. The differential represents the amount that the tangent line rises or falls, whereas represents the amount that the curve rises or falls when changes by an amount Since we have when is small. If we take then we have which says that the actual change in is approximately equal to the differential If is a known number and it is desired to calculate an approximate value for where is small, then

Definition (Differential) If is a differentiable function then the differential is defined by the equation where is an independent variable.

Example (Differential) Find the differential for the following functions.

(a) Find the differential for  

    Solution. For the derivative is



and so the differential of is



(b) Find the differential for .

    Solution. For the derivative is



and so the differential of is



(c) Find the differential for

    Solution. For the derivative is



and so the differential of is

Example (Approximations of Real Numbers) Use differentials to make approximations for the following real numbers.

(a) Approximate

    Solution. If then and using the linear approximation is,





(b) Approximate

    Solution. If then and using the linear approximation is,   



(c) Approximate   

    Solution. If then and using the linear approximation is,

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Differentials
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/differentials.html