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

Answer:

(i) The derivative of the function is $f' = \frac{1}{10}$.

(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:

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

Step-by-step explanation:

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

The derivative is defined by the following limit:

$f' = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ (1)

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

$f' = \lim_{h \to 0} \frac{\frac{1}{10}\cdot (x+h) - \frac{1}{3}-\frac{1}{10}\cdot x +\frac{1}{3}}{h}$

$f' = \lim_{h \to 0} \frac{\frac{1}{10}\cdot h }{h}$

$f' = \lim_{h \to 0} \frac{1}{10}$

$f' = \frac{1}{10}$

The derivative of the function is $f' = \frac{1}{10}$.

(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:

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