Question

Find the derivative of f(x) = 12x^2 + 8x at x = 9.

Answer 1

224

Explanation:

We will need the following rules for derivative:

`(f+g)'=f'+g'` Sum rule.

`(cf)'=cf'` Constant multiple rule.

`(x^n)'=nx^(n-1)` Power rule.

`(x)'=1` Slope of y=x is 1.

`f(x)=12x^2+8x`

`f'(x)=(12x^2+8x)'`

`f'(x)=(12x^2)'+(8x)'` by sum rule.

`f'(x)=12(x^2)+8(x)'` by constant multiple rule.

`f'(x)=12(2x)+8(1)` by power rule.

`f'(x)=24x+8`

Now we need to find the derivative function evaluated at x=9.

`f'(9)=24(9)+8`

`f'(9)=216+8`

`f'(9)=224`

In case you wanted to use the formal definition of derivative:

`f'(x)=\lim_(h \rightarrow 0) (f(x+h)-f(x))/(h)`

Or the formal definition evaluated at x=a:

`f'(a)=\lim_(h \rightarrow 0) (f(a+h)-f(a))/(h)`

Let's use that a=9.

`f'(9)=\lim_(h \rightarrow 0) (f(9+h)-f(9))/(h)`

We need to find f(9+h) and f(9):

`f(9+h)=12(9+h)^2+8(9+h)`

`f(9+h)=12(9+h)(9+h)+72+8h`

`f(9+h)=12(81+18h+h^2)+72+8h`

(used foil or the formula (x+a)(x+a)=x^2+2ax+a^2)

`f(9+h)=972+216h+12h^2+72+8h`

Combine like terms:

`f(9+h)=1044+224h+12h^2`

`f(9)=12(9)^2+8(9)`

`f(9)=12(81)+72`

`f(9)=972+72`

`f(9)=1044`

Ok now back to our definition:

`f'(9)=\lim_(h \rightarrow 0) (f(9+h)-f(9))/(h)`

`f'(9)=\lim_(h \rightarrow 0) (1044+224h+12h^2-1044)/(h)`

Simplify by doing 1044-1044:

`f'(9)=\lim_(h \rightarrow 0) (224h+12h^2)/(h)`

Each term has a factor of h so divide top and bottom by h:

`f'(9)=\lim_(h \rightarrow 0) (224+12h)/(1)`

`f'(9)=\lim_(h \rightarrow 0)(224+12h)`

`f'(9)=224+12(0)`

`f'(9)=224+0`

`f'(9)=224`