given (f(x)=7x + 9), find (f(x)) using the limit definition of the derivative.\n\n(f(x)=)

Explanation:

Step1: Recall the limit - definition of the derivative

The limit - definition of the derivative is $f^{\prime}(x)=\lim_{h\rightarrow0}\frac{f(x + h)-f(x)}{h}$. Given $f(x)=7x + 9$, then $f(x + h)=7(x + h)+9=7x+7h + 9$.

Step2: Substitute $f(x + h)$ and $f(x)$ into the formula

$\frac{f(x + h)-f(x)}{h}=\frac{(7x+7h + 9)-(7x + 9)}{h}$. Simplify the numerator: $(7x+7h + 9)-(7x + 9)=7x+7h + 9 - 7x - 9=7h$. So, $\frac{f(x + h)-f(x)}{h}=\frac{7h}{h}=7$.

Step3: Find the limit as $h\rightarrow0$

$f^{\prime}(x)=\lim_{h\rightarrow0}\frac{f(x + h)-f(x)}{h}=\lim_{h\rightarrow0}7$. Since the function $y = 7$ is a constant function, $\lim_{h\rightarrow0}7 = 7$.

Answer:

$7$