Hi. I have a question about a problem. It is: Calculate the derivative of each function directly from the definition. f(x) equals 1/(2x+1). The answers say it is -2/(2x+1)squared. I understand everything except how to get the correct denominator. Any help would be great!

Question

anonymous · Answer

whew... by the definition...

anonymous · Answer

When they say directly from the definition, they mean evaluate the following: $$
\lim_{h\to0}\frac{f(x+h)-f(x)}{h}.
$$So, we plug in $f(x)$, and obtain:$$
\lim_{h\to0}\frac{\frac{1}{2(x+h)+1}-\frac{1}{2x+1}}{h}.
$$Now we simplify this:$$
\lim_{h\to0}\frac{\frac{1}{2x+2h+1}-\frac{1}{2x+1}}{h}=\lim_{h\to0}\frac{\frac{2x+1-(2x+2h+1)}{(2x+2h+1)(2x+1)}}{h}=\lim_{h\to0}\frac{\frac{-2h}{(2x+2h+1)(2x+1)}}{h}\
=\lim_{h\to0}\frac{-2}{(2x+2h+1)(2x+1)}=\frac{-2}{(2x+1)(2x+1)}=\frac{-2}{(2x+1)^2}$$

anonymous · Answer

Sorry, had made a typo. Do you follow?

anonymous · Answer

Yeah except I do not understand how the denominator simplifies to (2x+1(2x+1).

anonymous · Answer

just plug in h=0 you end up with (2x+0+1) = (2x+1)

anonymous · Answer

Oh h=0...I kept trying to multiply it out and it wasn't giving me the right answer (obviously). I guess I just spaced out. Thanks I appreciate it!