# Thread: Derivatives of Inv. Trig

1. ## Derivatives of Inv. Trig

Hello everyone. I need help on the following problem involving inverse trig.

A) Show that the derivative of ( arccot[x] - arctan[1/x] ) = 0 for all x not equal to 0.

B) Prove that there is no constant C such that arccot[x] - arctan[1/x] = C for all x not equal to 0. Explain why this does not contradict the zero-derivative theorem.

Any suggestions will be appreciated

2. Originally Posted by RB06
Hello everyone. I need help on the following problem involving inverse trig.

A) Show that the derivative of ( arccot[x] - arctan[1/x] ) = 0 for all x not equal to 0.

B) Prove that there is no constant C such that arccot[x] - arctan[1/x] = C for all x not equal to 0. Explain why this does not contradict the zero-derivative theorem.

Any suggestions will be appreciated
Let x = tan(y), then 1/x = cot(y). Hence (assuming that atan and acot
return values in the range (-pi/2, pi/2):

y= atan(x) = acot(1/x).

So: acot(x) - atan(1/x) = 0, and so the derivative is zero (the derivative of a constant function is zero).

Also if atan abd acos have the restriction on range we have also proven part B.

RonL

3. Hello, RB06!

A) Show that the derivative of: .f(x) .= .arccot(x) - arctan(1/x) .
. . .equals 0 for all x ≠ 0.

We have: .f(x) .= .arccot(x) - arctan(x^{-1})

. . . . . . . .-1 . . . . . -x^{-2}
f'(x) .= . -------- - ---------------
. . . . . . .1 + x² . .1 + x^{-2}

Multiply top and bottom of the second fraction by x²:

. . . . . . . .-1 . . . . . 1
f'(x) .= .-------- + -------- . = . 0
. . . . . . 1 + x² - -x² + 1