# Thread: The Derivative as a unction

1. ## The Derivative as a unction

If f(t)= √((t^2-1)/(t^2-1))
Find f '(t)
I have tried it several times using f(x+h)-f(x) / h, but I always get lost and don't know how to solve it.
Could someone help me?
Thank you

2. Originally Posted by Raimuna
If f(t)= √((t^2-1)/(t^2-1))
Find f '(t)
I have tried it several times using f(x+h)-f(x) / h, but I always get lost and don't know how to solve it.
Could someone help me?
Thank you
are you sure it is $\displaystyle f(t)=\sqrt {\frac{t^2-1}{t^2-1} }$ ?

3. Originally Posted by Raimuna
If f(t)= √((t^2-1)/(t^2-1))
Find f '(t)
I have tried it several times using f(x+h)-f(x) / h, but I always get lost and don't know how to solve it.
Could someone help me?
Thank you
That's the function?

$\displaystyle f(t)=\sqrt{\frac{t^2-1}{t^2-1}}=1$

??

It's probably this:

$\displaystyle f(t)=\frac{\sqrt{t^2-1}}{t^2-1}$

right?

4. Originally Posted by ramiee2010
are you sure it is $\displaystyle f(t)=\sqrt {\frac{t^2-1}{t^2-1} }$ ?
sory, my bad. the denominator should be t^2+1

5. So we want to evaluate the limit:

$\displaystyle \lim_{h->0}(\frac{\sqrt{(t+h)^2-1}}{h(t+h)^2-h}-\frac{\sqrt{t^2-1}}{ht^2-h})$

6. Why dont you just use the rules of differntiation?
I dont see why you would try to find the derivative of f using the limit definition of the derivative.

Two obvious approaches that stand out.

first you could use the chain rule followed by the quotient rule for differentiation.

or simplify

$\displaystyle f(t)=\sqrt {\frac{t^2-1}{t^2 + 1} } = \frac{\sqrt {t^2-1}} {\sqrt{t^2 + 1}}$

then use the quotient rule and chain rule.

I find after some quick and sloppy calculations (so forgive me if this is not completely accurate)

$\displaystyle f'(t) = \frac{2t}{(t^2 - 1)(t^2 + 1)^{3/2}}$