1. ## differenciate

Find the derivative of the function y= x sin (1/x)

Differentiate: y= (u^6 - 2u^3 + 5) /u^2

Thanks!

2. Originally Posted by new yorker
Find the derivative of the function y= x sin (1/x)

Differentiate: y= (u^6 - 2u^3 + 5) /u^2

Thanks!
The first one is a combination of product and chain rule:

$y'=\sin\left(\frac{1}{x}\right)+x\cos\left(\frac{1 }{x}\right)\cdot\left(-\frac{1}{x^2}\right)=\color{red}\boxed{\sin\left(\ frac{1}{x}\right)-\frac{1}{x}\cos\left(\frac{1}{x}\right)}$

Does this make sense?

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The second one isn't bad:

$y=\frac{u^6-2u^3+5}{u^2}=\frac{u^6}{u^2}-2\frac{u^3}{u^2}+\frac{5}{u^2}=u^4-2u+\frac{5}{u^2}$

Can you differentiate this one?

--Chris

3. For the first question, use the product rule:

$y = x \sin \left(\frac{1}{x}\right)$

$\frac{dy}{dx} = x \left(-\frac{1}{x^2} \cdot \cos \left(\frac{1}{x}\right)\right) + \sin \left(\frac{1}{x}\right)$

For the second question, divide through by $u^2$:

$y = \frac{u^6 - 2u^3 + 5}{u^2}$

$y = u^4 - 2u + 5u^{-2}$

$\frac{dy}{dx} = 4u^3 - 2 - 10u^{-3}$