Tool to calculate the domain of definition of a function f(x), ie. the set of values x which exists through the derivative f'(x).

Domain of Derivative of a Function - dCode

Tag(s) : Functions

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The domain of derivability of a function is the domain of definition of the derivative of the function.

Calculating the derivation domain of a function, noted $ D_{f'} $, is to calculate the set of definition of its derivative function. Check in $ \mathbb{R} = ] -\infty ; +\infty [ $, the values for which the derivative function is not defined. That is, the values of $ x $ such that $ f'(x) $ does not exist.

The calculation of the derivation domain is thus composed of 2 steps:

Step 1: Calculate the derivative of the function

Step 2: Calculate the definition domain of the derivative calculated at step 1

__Example:__ $ f(x) = \ln(x) = \log(x) $ is defined over $ \mathbb{R}^{*+} = ] 0 ; +\infty [ $, its derivative is $ f'(x) = \frac{1}{x} $. Which definition domain is $ D_{f'} = \mathbb{R}^* = ] - \infty ; 0 [ \cup ] 0 ; +\infty [ $

A rational function of the form $ f(x) = \frac{P(x)}{Q(x)} $ has the same definition domain as its derivative. So any rational function is derivable on its own domain of definition.

Indeed, the derivative $ f'(x) = \frac{ P'(x)Q(x) - P(x)Q'(x) }{ Q(x)^2 } $ does not modify its domain of definition.

A function is differentiable only on the set of values where it is continuous, and therefore, it is continuous only on the values where it is defined.

Thus, the domain of derivability of a function is a subset of its domain of definition.

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