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

2 Answers
Jan 30, 2018

#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#

Jan 30, 2018

#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#