How do you use the Product Rule to find the derivative of # y = xsqrt(1-x^2)#?

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Answer 1 · 2015-08-16T19:45:19

#color(red)(dy/dx= sqrt(1-x^2)-x^2/sqrt(1-x^2))#

#y=x(1-x^2)^(1/2)#

The Product Rule, states that if #y=uv#, then

#dy/dx = u(dv)/dx+v(du)/dx#

Let #u=x# and #v=(1-x^2)^(1/2)#

Then

#(du)/dx=1#

#(dv)/dx=1/2(1-x^2)^(-1/2)(-2x)= -x(1-x^2)^(-1/2)#

#dy/dx = u(dv)/dx+v(du)/dx#

#dy/dx=-x^2(1-x^2)^(-1/2) +(1-x^2)^(1/2)#

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