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Question Number 28280 by Cheyboy last updated on 23/Jan/18

$$Finddy/dx$$$$x^{\frac{2}{3}}{(6-x)}^{\frac{1}{3}}toitsimplestform$$

Commented by abdo imad last updated on 23/Jan/18

$$wehavey\left(x\right)={\left(x^2(6-x)\right)}^{\frac{1}{3}}={(-x^3+6x^2)}^{\frac{1}{3}}so$$$$\frac{dy}{dx}\left(x\right)=\frac{1}{3}{(-3x^2+12x)}^{-\frac{2}{3}}.$$

Commented by Cheyboy last updated on 23/Jan/18

$$sirdbookishavingdifferentthing$$$$$$

Commented by abdo imad last updated on 23/Jan/18

$$trustyourselfsirbecausealotsofbooksarefullwitherror$$$$andperhapsthebookhavegivenanotherformofderivative$$$$equaltothismyansweriscorrect...$$

Commented by Cheyboy last updated on 24/Jan/18

$$okthankzsir$$$$$$

Answered by ajfour last updated on 24/Jan/18

$$y=x^{2/3}{(6-x)}^{1/3}$$$$\ln y=\frac{2}{3}\ln x+\frac{1}{3}\ln (6-x)$$$$\frac{1}{y}\frac{dy}{dx}=\frac{2}{3x}-\frac{1}{3(6-x)}$$$$\frac{dy}{dx}=y\left[\frac{12-2x-x}{3x(6-x)}\right]$$$$\frac{dy}{dx}=\frac{x^{2/3}{(6-x)}^{1/3}(4-x)}{x(6-x)}$$$$=\frac{4-x}{x^{1/3}{(6-x)}^{2/3}}.$$

Commented by Cheyboy last updated on 24/Jan/18

$$Exactlythatzwhatzinthebook$$

Commented by Cheyboy last updated on 24/Jan/18

$$Godblessusir$$

Commented by abdo imad last updated on 24/Jan/18

$$forthisderivativenoneedtouseln..$$

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