sqrt(1+x)/sqrt(1-x)
_______ \/ 1 + x --------- _______ \/ 1 - x
/ _______\ d |\/ 1 + x | --|---------| dx| _______| \\/ 1 - x /
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.
Apply the power rule: goes to
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:
_______ \/ 1 + x 1 ------------ + --------------------- 3/2 _______ _______ 2*(1 - x) 2*\/ 1 + x *\/ 1 - x
_______ 1 2 3*\/ 1 + x - ---------- + ----------------- + ----------- 3/2 _______ 2 (1 + x) \/ 1 + x *(1 - x) (1 - x) ---------------------------------------------- _______ 4*\/ 1 - x
/ _______\ | 1 1 3 5*\/ 1 + x | 3*|---------- - ------------------ + ------------------ + -----------| | 5/2 3/2 _______ 2 3 | \(1 + x) (1 + x) *(1 - x) \/ 1 + x *(1 - x) (1 - x) / ---------------------------------------------------------------------- _______ 8*\/ 1 - x