Question

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

Answer 1

$g^{\prime}(2)=-\frac{1}{5}$

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.

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

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

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

Hence, the answer is:

$g^{\prime}(2)=-\frac{1}{5}$