A function g is defined for all real numbers and has the following property:
g(a+b)-g(a)= 4ab+2b^2
Find g'(x)

Question

A function g is defined for all real numbers and has the following property:
g(a+b)-g(a)= 4ab+2b^2 
Find g'(x)

hartnn · Answer

do you know the first principle definition of derivative ??

anonymous · Answer

no...

hartnn · Answer

$\huge f'(x)=\lim \limits_{h \rightarrow 0}\dfrac{f(x+h)-f(x)}{h}$
haven't seen this ?

hartnn · Answer

thats the definition.

hartnn · Answer

so, for your question, first find , [g(a+b)-g(a)] /b =.... ?
can you ?

anonymous · Answer

do I factor out b from the right and divided everything by b?

hartnn · Answer

yes! do that , and tell me what you get ?

anonymous · Answer

$$\frac{ g(a+b)-g(a) }{b }=4a+2b$$

hartnn · Answer

correct !
but you want, 
 $\huge g'(x)=\lim \limits_{h \rightarrow 0}\dfrac{g(x+h)-g(x)}{h}=....$
right ?
so, just replace 'a' by 'x' and 'b' by 'h' in your last equation, what you get ?

anonymous · Answer

$$\frac{ g(x=h)-g(x) }{ h }=4x+2h??$$

hartnn · Answer

4x+2h is correct!
$\large g'(x)=\lim \limits_{h \rightarrow 0}\dfrac{g(x+h)-g(x)}{h}=\lim \limits_{h \rightarrow 0} (4x+2h)=.....?$
can you solve that limit ?

hartnn · Answer

just plug in h=0 there!

anonymous · Answer

4x?

hartnn · Answer

yes, $\large g'(x)=4x$
is correct final answer :) Ask if any doubts.

anonymous · Answer

oh I get it actually thank you!

hartnn · Answer

welcome ^_^