# What is the derivative of this function y = x sin (5/x)?

May 16, 2017

$\frac{\mathrm{dy}}{\mathrm{dx}} = \sin \left(\frac{5}{x}\right) - \frac{5}{x} \cos \left(\frac{5}{x}\right)$

#### Explanation:

First we need the product rule, which states that $\frac{d}{\mathrm{dx}} \left(u v\right) = \left(\frac{\mathrm{du}}{\mathrm{dx}}\right) v + u \left(\frac{\mathrm{dv}}{\mathrm{dx}}\right)$. Thus:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \left(\frac{d}{\mathrm{dx}} x\right) \sin \left(\frac{5}{x}\right) + x \left(\frac{d}{\mathrm{dx}} \sin \left(\frac{5}{x}\right)\right)$

Here, $\frac{d}{\mathrm{dx}} x = 1$.

To figure out $\frac{d}{\mathrm{dx}} \sin \left(\frac{5}{x}\right)$, we need the chain rule since we have a function inside another function. Knowing that $\frac{d}{\mathrm{dx}} \sin \left(x\right) = \cos \left(x\right)$, we see that through the chain rule, $\frac{d}{\mathrm{dx}} \sin \left(u\right) = \cos \left(u\right) \left(\frac{\mathrm{du}}{\mathrm{dx}}\right)$.

Then:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \sin \left(\frac{5}{x}\right) + x \cos \left(\frac{5}{x}\right) \left(\frac{d}{\mathrm{dx}} \left(\frac{5}{x}\right)\right)$

Note that $\frac{d}{\mathrm{dx}} \frac{5}{x} = \frac{d}{\mathrm{dx}} 5 {x}^{-} 1 = - 5 {x}^{-} 2$ through the power rule:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \sin \left(\frac{5}{x}\right) + x \cos \left(\frac{5}{x}\right) \cdot \left(- \frac{5}{x} ^ 2\right)$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \sin \left(\frac{5}{x}\right) - \frac{5}{x} \cos \left(\frac{5}{x}\right)$