Question

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 2 x − 1 9

Answer 1

If $f(x)=12x-19$ (whose domain is all real $x$), then the derivative is

$f'(x):=\displaystyle\lim_{h\to0}\frac{f(x+h)-f(x)}h$

$f'(x)=\displaystyle\lim_{h\to0}\frac{12(x+h)-19-(12x-19)}h=\lim_{h\to0}\frac{12h}h=\lim_{h\to0}12=12$

and similarly the domain of $f'(x)$ is all real $x$.

Please clarify your question if this is not the right interpretation of $f(x)$.