Differentiation question a level edexcel

I don’t think differentiating arc sin/cos/tan is part of the spec anymore ? I think the only question u can get is a show that question with just arc sin/cos/tan x

I think the proof is for example with arc sin, just show that the derivative of arc sin x = 1/root 1-x^2. You let arc sin = y so that x = sin y. Then dx/dy is cos y, so dy/dx is 1/cos y. They using sin^2y + cos^2y =1, rearrange in terms of cos y. So cos y = root of 1-sin^2y, and we know sin y=x, so cos y = root of 1-x^2. So dy/dx = 1/root 1-x^2.
Same sort of idea for arc cos and arc tan as well.

I understand that bit but I don’t think I showed it in my working out but I don’t understand where I went wrong in the above question. Could you help please?

I understand that bit but I don’t think I showed it in my working out but I don’t understand where I went wrong in the above question. Could you help please?

Your working is largely right, though the bit in the bottom right seems to give the reciprocal of the answer, so 1/sqrt(1+x^2). Dividing by a fraction is flip and multiply, or multiply through (num and denom) by sqrt(1+x^2)

Ohhh yeah so that would make it root 1+x^2 which would make dy by dx 1/1+x^2. Thank you for your help! Also would questions like this come up because the other person said they don’t think it’s part of the spec?

If you had
theta = arcsin(x/sqrt(1+x^2))
could you sketch that as a right triangle with sides .... and then get simpler arctrig function to differentiate? It may not be the way they want you to go, but the question (arcsin argument) is kind of hinting at it. It would be a few lines or a write down if you assume you know the arctrig derviative.

I’m sort of confused by this. I got a triangle with the sides 1 and x and the hyp being root 1+x^2 but I’m not really sure what you’re asking me to do.

So the right triangle has sides
adj: opp:hyp = 1:x:sqrt(1+x^2)
where theta = arcsin(x/sqrt(1+x^2)) = arctan(x)
so differentiating arcsin(x/sqrt(1+x^2)) gives the same as differentiating arctan(x)

But if the question was 8 marks would I get all the marks for doing that method?

But if the question was 8 marks would I get all the marks for doing that method?

But if the question was 8 marks would I get all the marks for doing that method?

Purely or interest, and to prove that you can do anything with a triangle, in Jim Belks ans in
https://math.stackexchange.com/questions/1299682/geometric-intuition-for-derivatives-of-basic-trig-functions
for the image with the main triangle which is 1:tan(theta):sec(theta) which is our scenario with tan(theta)=x, sec(theta)=sqrt(1+x^2), then the top small triangle corresponds to our scenario
https://i.sstatic.net/Zkwjz.png
so the dx corresponds to the small increment in x or tan(theta) and d theta is the small increment in theta
dx ~ sec^2(theta) d theta
or
d theta / dx ~ 1/(1+tan^2(theta)) = 1/(1+x^2)
Obviously, you need to take limiting arguments and its not an a level ans, but a simple picture of 3 triangles replaces a page of calculus/algebra.