# Differntiable at point problem

1. Aug 23, 2009

### james.farrow

f(x) = x/(3x + 1), prove f(x) is differentiable at point 2.

Ok so I've had several attempts at this....

Using Q(h) = (f(h) - f(2))/h

I eventually end up with (h^2 -2h)/(7(3h + 1))

Obviously the above is rubbish because it I differentiate f(x) using the normal rules then

dy/dx = 1/(3x + 1)^2

What am I missing here?

Also I've tried using the difference quotient to prove it is differentiable but same result - just rubbish.

f(c + h) - f(c)/ h

The above also doesn't work out either! Please hep!!

2. Aug 23, 2009

### jeffreydk

So you have the function $f(x)=\frac{x}{3x+1}$ and you know that a function is differentiable at $a$ if its derivative exists at $a$. You also know that

$$\left.\frac{df}{dx}\right|_{a}\equiv \lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h}=\frac{\frac{a+h}{3(a+h)+1}-\frac{a}{3a+1}}{h}$$

If you simplify this does it match what you expected by using the rules you know? When you take the limit does that prove the limit exists for $a=2$?

3. Aug 23, 2009

### HallsofIvy

Staff Emeritus
It would be better to use Q(h)= (f(2+ h)- f(2))/h!

Yes, because you used the wrong formula for the difference quotient.

4. Aug 23, 2009

### james.farrow

Jefferydk if I simplify it I end up with nothing like? Whats going on.....??????

5. Aug 23, 2009

### james.farrow

Err after a cup of tea and a break the penny drops......

Thanks for your help lads! No doubt I'll call on you again!!

James

6. Aug 23, 2009

### HallsofIvy

Staff Emeritus
A cup of tea works wonders!

(I used to do my best work after a couple glasses of rum- but then I could never find the papers where I had written it all down!)