AQA C3 What type of differentiation topic would this type of question be?

Hi, I did a past paper the AQA C3 June 2012 one and I did everything fine apart from when I got to the final question and I didn't have a clue what to do.

The questions which I didn't understand are c and all of d.
Is there a specific name for this type of differentiation with stuff on the bottom, and differentiating tan^-1(ax-b)
as well as differentiating to find the second derivative when it is a fraction like the dy/dx in C.

If there isn't a name for the topic does anyone know where I can find some additional resources with more questions which I can practise on? I don't know how to search for these types of questions as its the more advanced differentiation of C3 which I can't do.

this is prelude of implicit differentiation which will eventually cover in your board in C4

In C3 you laern that

if x = f(y), then dy/dx = 1 / dx/dy

So these questions are both 1/dx/dy questions, still for d(ii) I wouldn't have a clue how to differentiate something like that i've never seen it before

For d)ii)
You simply need to differentiate:
$\dfrac{1}{1+(x-1)^2}-\dfrac{d}{dx} \left(\dfrac{d}{dx}(\ln(x)) \right)$

As for the derivative of $\arctan(x)$, you should calculate the derivative at least once using the method TeeEm suggested, but from then on, I would consider it a quotable result, and I imagine it's in your formula book.

Yeah, I have to differentiate that, but I don't know how to, I can differentiate using chain rule, product rule, quotient rule, but for stuff with stuff squared underneath in a fraction and another fraction I haven't learnt how to differentiate that, do you have a guide or something I could use.
I can't do TeeEm's method on y=tan^-1(x-1)
I ended up with dy/dx = 1/sec^2 y
when I use it from the formula booklet I get the correct answer though

Yeah, I have to differentiate that, but I don't know how to, I can differentiate using chain rule, product rule, quotient rule, but for stuff with stuff squared underneath in a fraction and another fraction I haven't learnt how to differentiate that, do you have a guide or something I could use.
I can't do TeeEm's method on y=tan^-1(x-1)
I ended up with dy/dx = 1/sec^2 y
when I use it from the formula booklet I get the correct answer though

Yes and you found that sec^2y is x^2-2x+2 so dy/dx=1/x^2-2x+2

Look in your formula booklet: it should say that $\displaystyle \frac{\mathrm{d}}{\mathrm{d}x} \tan^{-1}x = \frac{1}{1+x^2}$ - so in your case, just replace the $x$ with $x-1$.

Ah right yeah I get that part now thanks



I managed to do that now, stuck with d(ii) only now

For part di) I got x=2 and x=1 for part dii) do I need to use quotient rule

You don't HAVE to use the quotient rule, but you may as well.

Well, you need to evalute $\frac{d}{dx} (\arctan x - \ln x)$ then once you've got that, differentiate whatever you have again - what do you get?

I'm trying to do the quotient rule for it, I get
u=1
v= x^2 -2x + 2
du/dx = 0
dv/dx = 2x-2

(x^2-2x+2)(0) - (1)(2x-2) / (x^2-2x+2)^2

= -2x+2 / (x^2-2x+2)^2

then for the 1/x i get x^-2

so overall its
-2x+2/(x^2-2x+2)^2 + x^-2

is that right

What other methods are there in order to differentiate a quotient?

This is simply the chain rule with $(x^2 - blah blah)^{-1}$

Almost d/dx(1/x) = -x^(-2) not +

Thanks, i got it now