What is the derivative of #x * ((4-x^2)^(1/2))#?

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Answer 1 · 2017-12-14T22:55:09

#-(2*(-2+x^2))/sqrt(4-x^2)#

#f'(a*b)=b*f'(a)+a*f'(b)#

Let #color(white)(88)a=x#

Let #color(white)(88)b=(4-x^2)^(1/2)#

#f'(a)=1#

#f'(b)=1/2(4-x^2)^(-1/2)*-2x=-x(4-x^2)^(-1/2)#

#f'(a*b)=(4-x^2)^(1/2) * 1+x * -x(4-x^2)^(-1/2)#

#->=(4-x^2)^(1/2)+(-x^2)/(4-x^2)^(1/2)=-(2*(-2+x^2))/sqrt(4-x^2)#