Turn on thread page Beta
    • Thread Starter
    Offline

    0
    ReputationRep:
    how would you go about differentiating

    x sin^-1 (x^2)

    the sin^-1 is an inverse

    im not rlly sure but i made sqrt(sin y) = x and sub it in, and differentiate as normal.

    my answer is:
    [sqrt(1-x^4) + x^2] / 2x
    • PS Helper
    Offline

    14
    PS Helper
    Do you know how to differentiate \sin^{-1} x? If not, working out how to do that on its own would be a good start. Then you can use the product rule and chain rule to complete the question.

    Spoiler:
    Show
    u = x, v = \sin^{-1} w, w = x^2\newline

\Rightarrow \dfrac{dy}{dx} = v\dfrac{du}{dx} + u\dfrac{dv}{dw}\dfrac{dw}{dx}
    Offline

    2
    ReputationRep:
    In the spoiler I've shown how to differentiate \sin^{-1}x.
    I did this (and perhaps I should have just given hints - in fact, I'll split my spoiler up into more spoilers - you should try to do it yourself if you can) because it's useful to know, and if you're doing A level maths, then you may need to reproduce it in an exam. You should know how to find the derivatives of \cos^{-1}x and \tan^{-1}x too.

    To find the derivative of \sin^{-1}(f(x)), apply the chain rule.
    Spoiler:
    Show

     y = \sin^{-1}x
     x = \siny
     1 = \cosy x \frac{dy}{dx}
     \frac{dy}{dx} = \frac{1}{\cosy}
     \sin^{2}y + \cos^{2}y = 1
    Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.
    \cosy = \sqrt{1 - (sin^{2}y)

     = \sqrt{1 - x^{2}}
    Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.
    \frac{d}{dx} (\sin^{-1}x) = \frac{1}{\sqrt(1 - x^{2})



    Then you can just do what Nuodai said.

    EDIT: It's so irritating that I spent quite a while typing out how to differentiate inverse sine in LaTeX, only to have it called potentially dangerous.
    I hate LaTeX.

    I wish it didn't look so pretty.

    Edit: WTF IS ERROR 2?
    AND WHAT IS ERROR 6?
    GOD DAMN

    Edit: Better. At least some of it works.

    Edit: I give up. Here it is:
    Spoiler:
    Show

    y = sin^-1x
    x = siny
    Spoiler:
    Show

    1 = cosy(dy/dx)
    (dy/dx) = 1/cosy
    Spoiler:
    Show

    sin^2y + cos^2y = 1
    Spoiler:
    Show

    cosy = \sqrt(1 - sin^2y)
    Spoiler:
    Show

    cosy = \sqrt(1 - x^2)
    (d/dx) sin^-1x = 1\[\sqrt(1-x^2)]
    Offline

    14
    (Original post by AnonyMatt)
    I hate LaTeX.
    You need to leave a space after commands like \sin and \cos.

    \sin ^{2}y + \cos ^{2}y = 1
     \cos y = \sqrt{1 - (sin ^2y)}
    You might want to justify this step further. At the moment it is unsatisfactory as a^2 = b \implies a = \pm \sqrt{b}. You have ignored the minus possibility.
    Offline

    2
    ReputationRep:
    (Original post by Kolya)
    You need to leave a space after commands like \sin and \cos.

    You might want to justify this step further. At the moment it is unsatisfactory as a^2 = b \implies a = \pm \sqrt{b}. You have ignored the minus possibility.
    Thanks!

    Hmm. Do you know, I've never actually noticed that the step is not justified. I think it's allowed to be written like that at A level.

    It's obvious to see that it must be positive, but that's not really a justification, is it? (Is it?! :confused: )

    Is it because
    Unparseable or potentially dangerous latex formula. Error 6: Image was not produced or one of its dimensions is too small.
    \cosy
    * is positive for  \frac{-\pi}{2}

< y < \frac{\pi}{2} ?

    EDIT: FML LATEX!!! * = cosy
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: January 24, 2010
Poll
Do you think parents should charge rent?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.