Differntiable at point problem

[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!

 

[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 $a=2$?

 

[HallsofIvy]

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

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

I eventually end up with (h^2 -2h)/(7(3h + 1))
Obviously the above is rubbish

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

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!

 

[james.farrow]

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

 

[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

 

[HallsofIvy]

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!)