Answer:
224
Step-by-step explanation:
We will need the following rules for derivative:
[tex](f+g)'=f'+g'[/tex] Sum rule.
[tex](cf)'=cf'[/tex] Constant multiple rule.
[tex](x^n)'=nx^{n-1}[/tex] Power rule.
[tex](x)'=1[/tex] Slope of y=x is 1.
[tex]f(x)=12x^2+8x[/tex]
[tex]f'(x)=(12x^2+8x)'[/tex]
[tex]f'(x)=(12x^2)'+(8x)'[/tex] by sum rule.
[tex]f'(x)=12(x^2)+8(x)'[/tex] by constant multiple rule.
[tex]f'(x)=12(2x)+8(1)[/tex] by power rule.
[tex]f'(x)=24x+8[/tex]
Now we need to find the derivative function evaluated at x=9.
[tex]f'(9)=24(9)+8[/tex]
[tex]f'(9)=216+8[/tex]
[tex]f'(9)=224[/tex]
In case you wanted to use the formal definition of derivative:
[tex]f'(x)=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}[/tex]
Or the formal definition evaluated at x=a:
[tex]f'(a)=\lim_{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}[/tex]
Let's use that a=9.
[tex]f'(9)=\lim_{h \rightarrow 0} \frac{f(9+h)-f(9)}{h}[/tex]
We need to find f(9+h) and f(9):
[tex]f(9+h)=12(9+h)^2+8(9+h)[/tex]
[tex]f(9+h)=12(9+h)(9+h)+72+8h[/tex]
[tex]f(9+h)=12(81+18h+h^2)+72+8h[/tex]
(used foil or the formula (x+a)(x+a)=x^2+2ax+a^2)
[tex]f(9+h)=972+216h+12h^2+72+8h[/tex]
Combine like terms:
[tex]f(9+h)=1044+224h+12h^2[/tex]
[tex]f(9)=12(9)^2+8(9)[/tex]
[tex]f(9)=12(81)+72[/tex]
[tex]f(9)=972+72[/tex]
[tex]f(9)=1044[/tex]
Ok now back to our definition:
[tex]f'(9)=\lim_{h \rightarrow 0} \frac{f(9+h)-f(9)}{h}[/tex]
[tex]f'(9)=\lim_{h \rightarrow 0} \frac{1044+224h+12h^2-1044}{h}[/tex]
Simplify by doing 1044-1044:
[tex]f'(9)=\lim_{h \rightarrow 0} \frac{224h+12h^2}{h}[/tex]
Each term has a factor of h so divide top and bottom by h:
[tex]f'(9)=\lim_{h \rightarrow 0} \frac{224+12h}{1}[/tex]
[tex]f'(9)=\lim_{h \rightarrow 0}(224+12h)[/tex]
[tex]f'(9)=224+12(0)[/tex]
[tex]f'(9)=224+0[/tex]
[tex]f'(9)=224[/tex]
