Okay so the question is:
Show that the function
2arccos(√a−xa−b)
is equal to
1√(a−x)(x−b).
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=−21√1−u2 du
See the Proof Wiki for a proof of this.
In this problem, we have, u=√a−xa−b, and we need to find dx, so we have:
ddx(√a−xa−b)=−√a−xa−b2(a−x)=−12√(a−b)(a−x)
So, lets put these two together.
ddu(2arccosu)=−21√1−u2 du=−2√1−(√a−xa−b)2(−12√(a−b)(a−x))
We can reduce this to:
ddx(2arccos(√a−xa−b))=1√(a−x)(x−b)
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