Dillard

2021-05-31

Find the derivatives.
$y={e}^{\frac{x}{3}}\cdot \left({x}^{6}-\frac{3}{x}\right)$

Ezra Herbert

The given function is $y={e}^{\frac{x}{3}}\cdot \left({x}^{6}-\frac{3}{x}\right)$
Obtain the derivative as follows.
$\frac{dy}{dx}=\frac{d}{dx}\left[{e}^{\frac{x}{3}}\left({x}^{6}-\frac{3}{x}\right)\right]$
$={e}^{\frac{x}{3}}\frac{d}{dx}\left({x}^{6}-\frac{3}{x}\right)+\left({x}^{6}-\frac{3}{x}\right)\frac{d}{dx}\left({e}^{\frac{x}{3}}\right)$
$={e}^{\frac{x}{3}}\left(6{x}^{5}+\frac{3}{{x}^{2}}\right)+\frac{1}{3}\left({x}^{6}-\frac{3}{x}\right){e}^{\frac{x}{3}}$
$={e}^{\frac{x}{3}}\left(\frac{1}{3}{x}^{6}+6{x}^{5}+\frac{3}{{x}^{2}}-\frac{1}{x}\right)$
Thus, $\frac{dy}{dx}={e}^{\frac{x}{3}}\left(\frac{1}{3}{x}^{6}+6{x}^{5}+\frac{3}{{x}^{2}}-\frac{1}{x}\right)$

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