Question

Use the inverse function-inverse cofunction

identities to derive the formula for the derivative of the function.
arccot(x)

Answer 1

`arccot(x)'=-(1)/(1+x^2)`

Explanation:

Use implicit differentiation:

`y=arccot(x)`

`cot(y)=x`

`(1)/(tan(y))=x`

`(cos(y))/(sin(y))=x`

`(dy)/(dx)((cos(y))/(sin(y)))=(dy)/(dx)(x)`

`(sin(y)(-sin(y))-cos(y)cos(y))/(sin^2(y))*(dy)/(dx)=1` (Quotient Rule:
`(d)/(dx)((f(x))/(g(x)))=(g(x)f'(x)-f(x)g'(x))/((g(x))^2)` )

`(-sin^2(y)-cos^2(y))/(sin^2(y))*(dy)/(dx)=1`

`(-(sin^2(y)+cos^2(y)))/(sin^2(y))*(dy)/(dx)=1`

`(-1)/(sin^2(y))*(dy)/(dx)=1`

`-csc^2(y)(dy)/(dx)=1`

`(dy)/(dx)=(1)/(-csc^2(y))`

`(dy)/(dx)=-sin^2(y)`

`(dy)/(dx)=-sin^2(arccot(x))`

See the attached picture to understand how to evaluate the mixed composition of trig functions using a right triangle.

Therefore, the derivative of arccot(x) is
`-(1)/(√(1+x^2))`.