Question

Find the derivative of the function using the definition of derivative. g(x) = 5 − x (1) Find g'(x) (2) State the domain of the function. (Enter your answer using interval notation.) (3) State the domain of its derivative. (Enter your answer using interval notation.)

Answer 1

Answer:

Answer is contained in the explanation

Step-by-step explanation:

$g(x)=5-x\\g'(x)=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\\g'(x)=\lim_{h \rightarrow 0} \frac{[5-(x+h)]-[5-x]}{h}\\g'(x)=\lim_{h \rightarrow 0} \frac{5-x-h-5+x}{h}\\g'(x)=\lim_{h \rightarrow 0} \frac{-h}{h}\\g'(x)=\lim_{h \rightarrow 0} -1\\g'(x)=-1$

g(x)=5-x has domain all real numbers (you can plug an a number and always get a number back)

So in interval notation this is $(-\infty, \infty)$

g'(x)=-1 has domain all real numbers (the original function had domain issues... and no matter the number you plug in you do get a number, that number being -1)

So in interval notation this is $(-\infty, \infty)$

Answer 2

The derivative of given function g(x) is

g'(x)=-1

Domain of function g(x) is (-∞,∞)

Domain of derivative is  (-∞,∞)

Given :

$g(x) = 5 - x$

Lets find derivative using definition of derivative

$\lim_{h \to 0} \frac{g(x+h)-g(x)}{h} \\g(x)=5-x\\g(x+h)=5-(x+h)\\g(x+h)=5-x-h\\\lim_{h \to 0} \frac{5-x-h-(5-x)}{h} \\\\\lim_{h \to 0} \frac{5-x-h-5+x}{h} \\\\\lim_{h \to 0} \frac{-h}{h} \\\\-1$

Derivative g'(x)=-1

g(x) is a linear function . for all linear function the domain is set of all real numbers

Domain of function g(x) is (-∞,∞)

Derivative function g'(x) =-1. For all values of x  the value of y is -1

So domain is set of all real numbers

Domain of derivative is  (-∞,∞)

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