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8.1 Derivatives of the Basic Functions

Since we define standard functions to be obtainable by applying a fixed set of operations (arithmetic ones, substitution and inversion) to combinations of three original functions, we can differentiate any standard function if we know how to differentiate each of the three functions, and how to differentiate a function obtained from others by each of the operations in terms of the derivatives of the others.

The derivatives of the three basic functions are as follows

x ' = 1 ( exp ⁡ x ) ' = exp ⁡ x sin ⁡ ' ( x ) = cos ⁡ x = sin ⁡ ( π 2 − x )

With rules for handling each operation, the task of differentiation of a standard function requires only parsing its definition to break it down into individual operations, and then applying the appropriate rule for each.

To do so we need rules for the following.

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