find the derivative of the function using the limit process.\n\n$f(x)=x^{2}+x - 9$\n\n$f(x)=lim_{delta…

find the derivative of the function using the limit process.\n\n$f(x)=x^{2}+x - 9$\n\n$f(x)=lim_{delta x\rightarrow0}\frac{f(x+delta x)-f(x)}{delta x}$\n\n$=lim_{delta x\rightarrow0}\frac{square-(x^{2}+x - 9)}{delta x}$\n\n$=square$

find the derivative of the function using the limit process.\n\n$f(x)=x^{2}+x - 9$\n\n$f(x)=lim_{delta x\rightarrow0}\frac{f(x+delta x)-f(x)}{delta x}$\n\n$=lim_{delta x\rightarrow0}\frac{square-(x^{2}+x - 9)}{delta x}$\n\n$=square$

Answer

Explanation:

Step1: Compute ( f(x+\Delta x) )

( f(x+\Delta x) = (x+\Delta x)^2 + (x+\Delta x) - 9 = x^2 + 2x\Delta x + (\Delta x)^2 + x + \Delta x - 9 )

Step2: Calculate ( f(x+\Delta x)-f(x) )

( [x^2 + 2x\Delta x + (\Delta x)^2 + x + \Delta x - 9] - (x^2 + x - 9) = 2x\Delta x + (\Delta x)^2 + \Delta x )

Step3: Divide by ( \Delta x )

( \frac{2x\Delta x + (\Delta x)^2 + \Delta x}{\Delta x} = 2x + \Delta x + 1 )

Step4: Take limit as ( \Delta x \to 0 )

( \lim_{\Delta x \to 0} (2x + \Delta x + 1) = 2x + 1 )

Answer:

( 2x + 1 )