How do you find the derivative of #(x)/sqrt(x^2-4)#?

1 Answer
Jim G.
Jan 8, 2017

#-4/(sqrt((x^2-4)^3)#

Explanation:

Differentiate using a combination of #color(blue)"quotient and chain rule"#

#"Given " f(x)=(g(x))/(h(x))" then"#

#color(red)(bar(ul(|color(white)(2/2)color(black)(f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2)color(white)(2/2)|)))larr"quotient rule"#

#g(x)=xrArrg'(x)=1#

#h(x)=sqrt(x^2-4)=(x^2-4)^(1/2)#

#rArrh'(x)=1/2(x^2-4)^(-1/2).d/dx(x^2-4)=x(x^2-4)^(-1/2)#

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

#=((x^2-4)^(-1/2)[x^2-4-x^2])/(x^2-4)#

#=(-4)/(x^2-4)^(3/2)=-4/(sqrt((x^2-4)^3)#

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