Question

find the derivative of $y = cos^{-1}left(\frac{1}{x}
ight)$ with respect to $x$.
the derivative of $y = cos^{-1}left(\frac{1}{x}
ight)$ with respect to $x$ is

Explanation:

Step1: Let $u=\frac{1}{x}$

$u = x^{-1}$

Step2: Differentiate $u$ with respect to $x$

$\frac{du}{dx}=-x^{-2}=-\frac{1}{x^{2}}$

Step3: Recall derivative of inverse - cosine function

The derivative of $y = \cos^{-1}(u)$ with respect to $u$ is $\frac{dy}{du}=-\frac{1}{\sqrt{1 - u^{2}}}$

Step4: Use chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$

Substitute $u=\frac{1}{x}$ into $\frac{dy}{du}$ and multiply by $\frac{du}{dx}$:
$\frac{dy}{dx}=-\frac{1}{\sqrt{1 - (\frac{1}{x})^{2}}}\cdot(-\frac{1}{x^{2}})$
$=\frac{1}{x^{2}\sqrt{1-\frac{1}{x^{2}}}}=\frac{1}{x^{2}\sqrt{\frac{x^{2}-1}{x^{2}}}}=\frac{1}{|x|\sqrt{x^{2}-1}}$ (for $|x|> 1$)

Answer:

$\frac{1}{|x|\sqrt{x^{2}-1}}$ (for $|x|>1$)