/ / __________\\ cot\asin\\/ -2*x + 2 //
d / / / __________\\\ --\cot\asin\\/ -2*x + 2 /// dx
There are multiple ways to do this derivative.
Rewrite the function to be differentiated:
Apply the quotient rule, which is:
and .
To find :
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Differentiate term by term:
The derivative of the constant is zero.
The derivative of a constant times a function is the constant times the derivative of the function.
Apply the power rule: goes to
So, the result is:
The result is:
The result of the chain rule is:
To find :
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Differentiate term by term:
The derivative of the constant is zero.
The derivative of a constant times a function is the constant times the derivative of the function.
Apply the power rule: goes to
So, the result is:
The result is:
The result of the chain rule is:
Now plug in to the quotient rule:
Rewrite the function to be differentiated:
Apply the quotient rule, which is:
and .
To find :
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Differentiate term by term:
The derivative of the constant is zero.
The derivative of a constant times a function is the constant times the derivative of the function.
Apply the power rule: goes to
So, the result is:
The result is:
The result of the chain rule is:
To find :
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Differentiate term by term:
The derivative of the constant is zero.
The derivative of a constant times a function is the constant times the derivative of the function.
Apply the power rule: goes to
So, the result is:
The result is:
The result of the chain rule is:
Now plug in to the quotient rule:
Now simplify:
The answer is:
/ 2/ / __________\\\ -\-1 - cot \asin\\/ -2*x + 2 /// --------------------------------- __________ __________ \/ -1 + 2*x *\/ -2*x + 2
___ / -1 + 2*x\ / 1 1 1 \ \/ 2 *|2 - --------|*|--------- - ------ - --------| \ -1 + x / \2*(1 - x) -1 + x -1 + 2*x/ ---------------------------------------------------- _______ __________ 4*\/ 1 - x *\/ -1 + 2*x
/ -1 + 2*x \ | 2 - -------- | ___ / -1 + 2*x\ | 3 6 7 2 6 -1 + x | \/ 2 *|2 - --------|*|--------- + ----------- + ---------- - ------------------ + ------------------- + ------------------| \ -1 + x / | 2 2 2 (1 - x)*(-1 + 2*x) (-1 + x)*(-1 + 2*x) (1 - x)*(-1 + 2*x)| \(-1 + x) (-1 + 2*x) 2*(1 - x) / --------------------------------------------------------------------------------------------------------------------------- _______ __________ 8*\/ 1 - x *\/ -1 + 2*x