derivative using the definiton of derivative

[coolman]

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivatives.

g(t) = 1 / ?t

i did lim g(x+h) -g(x)/h = [1/(?x+h) - 1/?t] / h.

i don't know what to do next
Any help is appreciated.
thanks!

 

[galactus]

This one may be a little tricky.

\(\displaystyle \frac{1}{(t+h)^{\frac{1}{2}}}-\frac{1}{t^{\frac{1}{2}}}=\frac{t^{\frac{1}{2}}-(t+h)^{\frac{1}{2}}}{t^{\frac{1}{2}}(t+h)^{\frac{1}{2}}}\)

That last method was too drawn out and too difficult.

This is easier.

Multiply top and bottom by \(\displaystyle t^{\frac{1}{2}}+(t+h)^{\frac{1}{2}}\)

\(\displaystyle \frac{(t^{\frac{1}{2}}-(t+h)^{\frac{1}{2}})}{ht^{\frac{1}{2}}(t+h)^{\frac{1}{2}}}\cdot\frac{(t^{\frac{1}{2}}+(t+h)^{\frac{1}{2}})}{(t^{\frac{1}{2}}+(t+h)^{\frac{1}{2}})}\)

and it whittles down to:

\(\displaystyle \frac{-\not{h}^{1}}{\not{h}(t(t+h)^{\frac{1}{2}}+t^{\frac{1}{2}}(t+h)}\)

Let h=0 and the derivative emerges.

 

[coolman]

thank you.