Question

find the derivative of y = arccos(7/x) with respect to x.
the derivative of y = arccos(7/x) with respect to x is -7 / (sqrt(1 - (7/x)^2) * (x^2))

Explanation:

Step1: Recall chain - rule and derivative of arccos(u)

The derivative of $y = \arccos(u)$ with respect to $x$ is $\frac{dy}{dx}=-\frac{1}{\sqrt{1 - u^{2}}}\cdot\frac{du}{dx}$ by the chain - rule. Here $u=\frac{7}{x}=7x^{-1}$.

Step2: Find the derivative of $u = 7x^{-1}$

Using the power rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, for $u = 7x^{-1}$, we have $\frac{du}{dx}=-7x^{-2}=-\frac{7}{x^{2}}$.

Step3: Substitute $u$ and $\frac{du}{dx}$ into the chain - rule formula

Substitute $u = \frac{7}{x}$ and $\frac{du}{dx}=-\frac{7}{x^{2}}$ into $\frac{dy}{dx}=-\frac{1}{\sqrt{1 - u^{2}}}\cdot\frac{du}{dx}$. Then $\frac{dy}{dx}=-\frac{1}{\sqrt{1-(\frac{7}{x})^{2}}}\cdot(-\frac{7}{x^{2}})=-\frac{7}{x^{2}\sqrt{1 - (\frac{7}{x})^{2}}}$.

Answer:

$-\frac{7}{x^{2}\sqrt{1 - (\frac{7}{x})^{2}}}$