Find the derivative of the function using the definition of derivative, \( f(x)=m x+b \) \( f^{\prime}(x)=\square \) State the domain of the function. (Enter your answer using interval notation.) State the domain of its derivative. (Enter your answer using interval notation.)

To find the derivative of the function \( f(x) = mx + b \) using the definition of the derivative, we start with the definition: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] First, we calculate \( f(x+h) \): \[ f(x+h) = m(x+h) + b = mx + mh + b \] Now, we can substitute \( f(x+h) \) and \( f(x) \) into the derivative formula: \[ f'(x) = \lim_{h \to 0} \frac{(mx + mh + b) - (mx + b)}{h} \] Simplifying the expression inside the limit: \[ f'(x) = \lim_{h \to 0} \frac{mx + mh + b - mx - b}{h} = \lim_{h \to 0} \frac{mh}{h} \] Since \( h \neq 0 \), we can cancel \( h \): \[ f'(x) = \lim_{h \to 0} m = m \] Thus, the derivative of the function is: \[ f'(x) = m \] ### Domain of the Function The function \( f(x) = mx + b \) is a linear function, which is defined for all real numbers. Therefore, the domain of the function is: \[ \text{Domain of } f(x): (-\infty, \infty) \] ### Domain of the Derivative The derivative \( f'(x) = m \) is a constant function, which is also defined for all real numbers. Therefore, the domain of the derivative is: \[ \text{Domain of } f'(x): (-\infty, \infty) \] ### Summary - The derivative is \( f'(x) = m \). - Domain of \( f(x) \): \( (-\infty, \infty) \) - Domain of \( f'(x) \): \( (-\infty, \infty) \)