Tuesday, May 31, 2016

calculus - Differentiation of 2arccosleft(sqrtfracaxabright)



Okay so the question is:




Show that the function
2arccos(axab)



is equal to
1(ax)(xb).




I started by changing the arccosine into inverse cosine, then attempted to apply chain rule but I didn't get very far.
Then I tried substituting the derivative for arccosine in and then applying chain rule. Is there another method besides chain rule I should use? Any help is appreciated.


Answer



ddu2arccosu=211u2 du



See the Proof Wiki for a proof of this.




In this problem, we have, u=axab, and we need to find dx, so we have:



ddx(axab)=axab2(ax)=12(ab)(ax)



So, lets put these two together.



ddu(2arccosu)=211u2 du=21(axab)2(12(ab)(ax))



We can reduce this to:




ddx(2arccos(axab))=1(ax)(xb)


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