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Chapter 2: Q. A (page 238)

Use the definition of the derivative to prove that the derivative of a sum of functions fx+gxis equal to the sum of their derivativesf'x+g'x

Short Answer

Expert verified

ddxfx+gx=f'x+g'x

Step by step solution

01

Step 1:Given information

The given expression isfx+gx

02

Step 2:Explanation

We鈥檒l start with the sum of two functions. First recall the sum into the definition of the derivative and rewrite the numerator a little.

fx+gx'=limh鈫�0fx+h+gx+h-fx+gxh

=limh鈫�0fx+h-fx+gx+h-gxh

Now, break up the fraction into two pieces and recall that the limit of a sum is the sum of the limits

鈬�fx+gx'=limh鈫�0fx+h-fxh+limh鈫�0gx+h-gxh

=f'x+g'x

The proof of the difference of two functions in nearly identical

fx-gx'=limh鈫�0fx+h-gx+h-fx-gxh

=limh鈫�0fx+h-fx-gx+h-gxh

鈬�fx-gx'=limh鈫�0fx+h-fxh-limh鈫�0gx+h-gxh

=f'x-g'x

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