Question

What is the derivative of $co{t}^{-1}\left(x\right)$?

Answer 1

Determine the derivative of $co{t}^{-1}\left(x\right)$.

Let us assume
$$\begin{gathered} y={\cot }^{-1}x \\ \Rightarrow \cot y=x..................\left(i\right) \end{gathered}$$

Differentiate Implicitly
$$\begin{gathered} -cose{c}^{2}y\frac{dy}{dx}=1 \\ \Rightarrow \frac{dy}{dx}=-\frac{1}{cose{c}^{2}y} \end{gathered}$$
By trigonometric identity, we know that $cose{c}^2\theta =1+{\cot }^2\theta$ using the identity in the above equation we get,

$$\begin{gathered} \frac{dy}{dx}=-\frac{1}{1+{\cot }^{2}y} \\ \Rightarrow =-\frac{1}{1+x^2}fromequation..\left(I\right) \end{gathered}$$
Hence, the derivative of $co{t}^{-1}x$ is $"-\frac{1}{1+x^2}"$.