This is a product of two functions. We have to use the product rule:
y=sqrt(x-1)(x+1)=f(x)g(x)y=√x−1(x+1)=f(x)g(x)
f(x)=sqrt(x-1)=(x-1)^(1/2)=(x-1)^(1/2)f(x)=√x−1=(x−1)12=(x−1)12
g(x)=(x+1)g(x)=(x+1)
f'(x)=1/2(x-1)^(1/2-1)=1/2(x-1)^(-1/2)(1)=1/(2(x-1)^(1/2))=1/(2sqrt(x-1))
g'(x)=1
dy/dx=f'(x)g(x)+g'(x)f(x)
dy/dx=1/(2sqrt(x-1))(x+1)+(1)(sqrt(x-1))
dy/dx=(x+1)/(2sqrt(x-1))+sqrt(x-1)
dy/dx=((x+1)+2sqrt(x-1)sqrt(x-1))/(2sqrt(x-1))
dy/dx=(x+1+2(x-1))/(2sqrt(x-1)
dy/dx=(x+1+2x-2)/(2sqrt(x-1))=(3x-1)/(2sqrt(x-1)
Now we can multiply both top and bottom by sqrt(x-1) to normalize it:
dy/dx=((3x-1)sqrt(x-1))/(2sqrt(x-1)sqrt(x-1))=((3x-1)sqrt(x-1))/(2(x-1))