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Domain of Derivative of a Function

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).

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Domain of Derivative of a Function -

Tag(s) : Functions

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# Domain of Derivative of a Function

## Derivative Domain Calculator

 Format Inégalité … < … Interval […;…]

### What is the domain of derivative of a function? (Definition)

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

### How to calculate the domain of derivative of a 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 $D_{f'} = \mathbb{R}^* = ] - \infty ; 0 [ \cup ] 0 ; +\infty [$

### What is the domain of derivability of a rational function?

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.

### What is the relationship between the domain of derivability and the 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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