Question

Let g(x) =arccot x. using observation about the derivative of co functions find g’(2).

Answer 1

`g^(\prime)(2)=-(1)/(5)`

1) Considering this function:

`g(x)=\text{\textrm{arccot(x)}}`

2) Let's find the first derivative of that function, using the chain rule, note that we had to rewrite it as the arctan(1/x) and then make use of the chain rule for that.

`\begin{gathered} \frac{\mathrm{d} }{\mathrm{d} x}(\text{\textrm{arccot}}(x))= \\ \frac{\mathrm{d}}{\mathrm{d}x}\lbrack arc\tan ((1)/(x))\rbrack= \\ (1)/(((1)/(x))^2+1)\cdot(d)/(dx)((1)/(x))= \\ -(1)/(((1)/(x^2)+1)\cdot x^2) \\ -(1)/(x^2+1) \end{gathered}`

2.2) Now, we can plug into that x=2:

`\begin{gathered} g^(\prime)(x)=-(1)/(x^2+1) \\ g^(\prime)(2)=-(1)/(4+1) \\ g^(\prime)(2)=-(1)/(5) \end{gathered}`

Hence, the answer is:

`g^(\prime)(2)=-(1)/(5)`