Question

Use the definition of the derivative to find f’ when f(x) =6x^2 +4. No points for any other methods please help??

Answer 1

`f'(x)=12x`

Explanation:

According to definition of derivative for `f(x)`,

`f'(x)=(df)/(dx)=lim_(deltax->0)(f(x+deltax)-f(x))/(deltax)`

Here we have `f(x)=6x^2+4`, therefore

`f(x+deltax)=6(x+deltax)^2+4`

and `f'(x)=(df)/(dx)=lim_(deltax->0)(6(x+deltax)^2+4-(6x^2+4))/(deltax)`

= `lim_(deltax->0)(6x^2+12xdeltax+6(deltax)^2+4-6x^2-4)/(deltax)`

= `lim_(deltax->0)(12xdeltax+6(deltax)^2)/(deltax)`

= `lim_(deltax->0)12x+6deltax`

= `12x`

Answer 2

`f'(x) = 12x`

Explanation:

For any function `f`, the derivative `f'`, is given by:

`f'(x) = lim_(hrarr0) (f(x+h) - f(x))/h`

For `f(x) = 6x^2 + 4`, this is:

`f'(x) = lim_(hrarr0) (6(x+h)^2 + 4 - (6x^2 +4))/h`
`=lim_(hrarr0) (6x^2 + 12xh +6h^2 +4 - 6x^2 -4)/h`
`= lim_(hrarr0) (12xh + 6h^2)/h`
`=lim_(hrarr0)12x + 6h`

Now it is just a matter of letting `h=0` which leaves:

`f'(x) = 12x`