Question

let y = x/sin(x). dy/dx =

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $\frac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^{2}}$. Here, $u = \sin(x)$ and $v=x$.

Step2: Find derivatives of $u$ and $v$

We know that $\frac{du}{dx}=\cos(x)$ (derivative of $\sin(x)$) and $\frac{dv}{dx} = 1$ (derivative of $x$).

Step3: Substitute into quotient - rule formula

$\frac{dy}{dx}=\frac{x\cos(x)-\sin(x)\times1}{x^{2}}=\frac{x\cos(x)-\sin(x)}{x^{2}}$

Answer:

$\frac{x\cos(x)-\sin(x)}{x^{2}}$