# What is the derivative of sin(pi*x)?

Oct 18, 2015

$\pi \cos \pi x$

#### Explanation:

$\frac{d}{\mathrm{dx}} \sin \pi x = \pi \cos \pi x$

Feb 17, 2017

$\frac{\mathrm{dy}}{\mathrm{dx}} \text{ "=" "pixxcos(u)" "=" } \pi \cos \left(\pi x\right)$

These days they prefer the notation:

${f}^{'} \left(x\right) = \pi \cos \left(\pi x\right)$ or

#### Explanation:

$\textcolor{b l u e}{\text{Step 1}}$

Set $u = \pi x$

$\pi$ is a constant so $\frac{\mathrm{du}}{\mathrm{dx}} = \pi$

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$\textcolor{b l u e}{\text{Step 2}}$

Set $y = \sin \left(\pi x\right) \text{ "->" } y = \sin \left(u\right)$

Just accept that $\frac{d}{\mathrm{du}} \left(\sin \left(u\right)\right) = \cos \left(u\right)$

So $\frac{\mathrm{dy}}{\mathrm{du}} = \cos \left(u\right)$

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$\textcolor{b l u e}{\text{Step 3}}$

However; $\text{ "(du)/dx xx dy/(du) " "=" " dy/dxxx(du)/(du)" " =" } \frac{\mathrm{dy}}{\mathrm{dx}}$

So by substitution we have:

$\frac{\mathrm{dy}}{\mathrm{dx}} \text{ "=" "pixxcos(u)" "=" } \pi \cos \left(\pi x\right)$