How do you differentiate # f(x)= xsqrt(1-x^2)# using the product rule?

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Answer 1 · 2015-12-25T02:19:42

#f'(x)=(1-2x^2)/sqrt(1-x^2)#

The product rule: #d/dx[g(x)h(x)]=g'(x)h(x)+g(x)h'(x)#

#color(white)(xx)f'(x)=d/dx[x] * sqrt(1-x^2)+xd/dx[sqrt(1-x^2)]#

Find each derivative separately.

#color(white)(xx)d/dx[x]=1#

Use the chain rule—recall that #d/dx[u^(1/2)]=1/2u^(-1/2) * (du)/dx#.

#color(white)(xx)d/dx[(1-x^2)^(1/2)]=1/2(1-x^2)^(-1/2) * d/dx[-x^2]#

#color(white)(xx)=>1/(2sqrt(1-x^2))(-2x)=-x/sqrt(1-x^2)#

Plug these back in:

#color(white)(xx)f'(x)=sqrt(1-x^2)-x^2/sqrt(1-x^2)#

#color(white)(xx)=>(1-2x^2)/sqrt(1-x^2)#