Question

find the derivative of the function using the definition of derivative. f(x) = mx + b f(x) = state the domain of the function. (enter your answer using interval notation.) state the domain of its derivative. (enter your answer using interval notation.) resources read it

Explanation:

Part 1: Find the derivative using the definition of derivative

Step 1: Recall the definition of the derivative

The definition of the derivative of a function \( f(x) \) is:
\[
f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
\]

Step 2: Substitute \( f(x + h) \) and \( f(x) \) into the definition

Given \( f(x) = mx + b \), we have:
\( f(x + h) = m(x + h) + b = mx + mh + b \)
So,
\[
\frac{f(x + h) - f(x)}{h} = \frac{(mx + mh + b) - (mx + b)}{h}
\]

Step 3: Simplify the numerator

Simplify the numerator:
\[
\frac{(mx + mh + b) - (mx + b)}{h} = \frac{mx + mh + b - mx - b}{h} = \frac{mh}{h}
\]

Step 4: Cancel \( h \) (for \( h

eq 0 \)) and take the limit
Cancel \( h \) (since \( h \to 0 \) but \( h
eq 0 \) in the limit process):
\[
\frac{mh}{h} = m
\]
Now, take the limit as \( h \to 0 \):
\[
\lim_{h \to 0} m = m
\]
So, \( f'(x) = m \)

Part 2: State the domain of the function \( f(x) = mx + b \)

The function \( f(x) = mx + b \) is a linear function. Linear functions are defined for all real numbers. In interval notation, the set of all real numbers is \( (-\infty, \infty) \)

Part 3: State the domain of its derivative \( f'(x) = m \)

The derivative \( f'(x) = m \) is a constant function. Constant functions are defined for all real numbers. In interval notation, this is \( (-\infty, \infty) \)

Answer:

s:

• Derivative: \( \boldsymbol{m} \)
• Domain of \( f(x) \): \( \boldsymbol{(-\infty, \infty)} \)
• Domain of \( f'(x) \): \( \boldsymbol{(-\infty, \infty)} \)