Question

Question #449c9

Answer 1

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

Explanation:

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))`