Differentiability

Source: Principles of Mathematical Analysis

Let . The derivative of at (if it exists) is given . If is defined at , it is differentiable at . Furthermore if is defined everywhere on , is differentiable on .

Continuity

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Let be defined on if is differentiable at , then is continuous at .

As , we have .

Arithmetic

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Suppose and are differentiable at . Then , , and are differentiable at with

3. (provided )

(1) Follows by sum of limits.

(2) Let . Then . Divide by and note as . Successive application of algebraic limit theorem gives the limit.

(3) Letting , then . As the desired limit follows.

Chain Rule

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Let be continuous on and defined for , while is defined on an interval containing the range of and is differentiable at . If for , then is differentiable at and .

Let , and take , and as , as such that the following holds.

Applying these substitutions and assuming , we have

Note gives , and this quantity converges to .