Question

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Find the derivative of the function using the definition of derivative.
State the domain of the function and the domain of its derivative.
f(x) = 1/10x-1/3

Answer 1

(i) The derivative of the function is
`f' = (1)/(10)`.

(ii) The domain of all first order polynomials (linear functions) is the set of all real numbers. That is:

`Dom\{f(x)\} = \mathbb{R}`

The domain of all zero order polynomials (constant functions) is the set of all real numbers. That is:

`Dom\{f'\} = \mathbb{R}`

Explanation:

(i) Find the derivative of the function using the definition of derivative:

The derivative is defined by the following limit:

`f' = \lim_(h \to 0) (f(x+h)-f(x))/(h)` (1)

If we know that
`f(x) = (1)/(10)\cdot x - (1)/(3)`, then the definition of derivative is expanded:

`f' = \lim_(h \to 0) ((1)/(10)\cdot (x+h) - (1)/(3)-(1)/(10)\cdot x +(1)/(3))/(h)`

`f' = \lim_(h \to 0) ((1)/(10)\cdot h )/(h)`

`f' = \lim_(h \to 0) (1)/(10)`

`f' = (1)/(10)`

The derivative of the function is
`f' = (1)/(10)`.

(ii) State the domain of the function and the domain of its derivative:

The domain of all first order polynomials (linear functions) is the set of all real numbers. That is:

`Dom\{f(x)\} = \mathbb{R}`

The domain of all zero order polynomials (constant functions) is the set of all real numbers. That is:

`Dom\{f'\} = \mathbb{R}`