Question #449c9

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Answer 1 · 2017-10-19T07:57:35

$dy/dx=sqrt(1-x^3)-(3x^3)/(2sqrt(1-x^3))$

We have $y=xsqrt(1-x^3)$

We will differentiate both sides with respect to $x$ using the chain rule and the product rule.

The product rule is $d/dx(U*V)=color(blue)Ud/dxcolor(red)V+color(blue)Vd/dxcolor(red)U$

$d/dxy=d/dx(xsqrt(1-x^3))$

$dy/dx=color(blue)sqrt(1-x^3)d/dxcolor(red)x+color(blue)xd/dxcolor(red)sqrt(1-x^3)d/dxcolor(green)(1-x^3)$

$dy/dx=color(blue)sqrt(1-x^3)*color(red)1+color(blue)xcolor(red)(1/(2sqrt(1-x^3))*color(green)((-3x^2))$

$dy/dx=sqrt(1-x^3)-(x3x^2)/(2sqrt(1-x^3))$

$dy/dx=sqrt(1-x^3)-(3x^3)/(2sqrt(1-x^3))$