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# Differential of inverse trig functions? watch

1. Does anyone have any good links to some websites/videos on how to differentiate sin/cos/tan^-1 x?

I could memorise the end products of each one but I would prefer to know how to derive them myself because I've seen questions come up such as;

Differentiate Tan^-1(5x) when first stating tan^-1x = tany Or something similar.

Any help?
2. Start by writing y = cos^-1 x
This is the same as cos y = x
Then you can differentiate implicitly with respect to x. Unfortunately I can't go further than that at this stage... see what you get!

Edit: Or try it with tan/sin in the same form, I just chose cosine because it's the best.
3. (Original post by Shawshank)
Start by writing y = cos^-1 x
This is the same as cos y = x
Then you can differentiate implicitly with respect to x. Unfortunately I can't go further than that at this stage... see what you get!

Edit: Or try it with tan/sin in the same form, I just chose cosine because it's the best.
Just adding onto this, there will come a stage in your working where you use the identities and .
4. Always start by reexpressing the equation without the inverse tirg function. for your example it would be like this

Now differentiate, using implicit differentiation on the LHS

Rearrange to get as subject

Final stage is to remove the y terms on the RHS. In this casr substituting in for using trig identity as from first part. Final result is therefore

Hope that is useful!
5. This is pretty similar, but isn't there a general formula?

In particular, since (using the formula for differentiating a fraction), , and then indeed one has to express via to get rid of .
6. (Original post by spread_logic_not_hate)
Always start by reexpressing the equation without the inverse tirg function. for your example it would be like this

Now differentiate, using implicit differentiation on the LHS

Rearrange to get as subject

Final stage is to remove the y terms on the RHS. In this casr substituting in for using trig identity as from first part. Final result is therefore

Hope that is useful!
That was incredibly usefull. Thanks, that's what I was looking for somebody to explain it step by step. I'm going to go try it for the other functions.
7. (Original post by RollerBall)
I'm going to go try it for the other functions.
it's really quite fun
8. (Original post by RollerBall)
Does anyone have any good links to some websites/videos on how to differentiate sin/cos/tan^-1 x?

I could memorise the end products of each one but I would prefer to know how to derive them myself because I've seen questions come up such as;

Differentiate Tan^-1(5x) when first stating tan^-1x = tany Or something similar.

Any help?
Is this Edexcel C3
9. (Original post by FZka)
Is this Edexcel C3
No it's Further Maths.
10. (Original post by Mathematician!)
No it's Further Maths.
I see. Thank you.
11. (Original post by spread_logic_not_hate)
Hope that is useful!
I tried doing it with

I did:

Now this is wrong. It should be

But this is according to my maths teacher who sometimes makes mistakes. But I'm sure I'm wrong. Any ideas?

EDIT: I realised what I've done wrong.

I should have done:

Therefore:

12. (Original post by RollerBall)
I tried doing it with

how did you go from cosy to 1-siny?
13. (Original post by rbnphlp)
how did you go from cosy to 1-siny?
Yeah, check my edit. I realised where I went wrong.
14. (Original post by RollerBall)
Yeah, check my edit. I realised where I went wrong.
15. (Original post by rbnphlp)
So Trig inverse functions you don't need to square root/square becuase they indirectly get squared when differentiating to sec^2x, but the others I need to square myself right?

So in theory, cos/sin should always have a square root in it, and tan shouldn't?
16. For

For

For

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