Question

Find g'(3) by using the definition of the derivative for the funcation g(u)=2/u+1

Answer 1

The definition of derivative of f(x) with respect to x is
\[f'(x) = \lim_{h \rightarrow 0} \frac{f(x + h) - f(x)}{h}\]

Answer 2

In this case, to find g'(3), first find g'(u):
\[g'(u) = \lim_{h \rightarrow 0} \dfrac{\dfrac{2}{(u + h) + 1} - \dfrac{2}{u + 1}}{h}\]

Answer 3

Next simplify the numerator:

\[\left(\frac{2}{u + h + 1}*\frac{u + 1}{u + 1}\right) - \left(\frac{2}{u + 1}*\frac{u + h + 1}{u + h + 1}\right)\]

Answer 4

thanks

Answer 5

We're not done yet

Answer 6

\[g'(u) = \lim_{h \rightarrow 0} \frac{2(u + 1) - 2(u + h + 1)}{h(u + 1)(u + h + 1)}\]\[g'(u) = \lim_{h \rightarrow 0} \frac{2u + 2 - 2u - 2h - 2}{h(u + 1)(u + h + 1)}\]\[g'(u) = \lim_{h \rightarrow 0} \frac{-2h}{h(u + 1)(u + h + 1)}\]
\[g'(u) = \frac{-2}{(u + 1)(u + 0 + 1)}\]
\[g'(u) = \frac{-2}{(u + 1)^2}\]

From here you should be able to evaulate \(g'(3)\)