The answer is: #y'=((5x-2)(x-2))/(2sqrt(x-3))#.
The rule of the derivative of a product is:
#y=f(x)*g(x)rArry'=f'(x)*g(x)+f(x)*g'(x)#;
so:
#y'=2xsqrt(x-3)+(x^2+4)*1/(2sqrt(x-3))=#
#=(2xsqrt(x-3)*2sqrt(x-3)+x^2+4)/(2sqrt(x-3))=(4x(x-3)+x^2+4)/(2sqrt(x-3))=#
#=(5x^2-12x+4)/(2sqrt(x-3))#
We can factor #5x^2-12x+4# remembering that if we consider that polynomial like a square equation, we can find the two solutions #x_1andx_2#, than:
#ax^2+bx+c=a(x-x_1)(x-x_2)#;
so, we can use the reduced formula because #b# is even:
#5x^2-12x+4=0rArr# #Delta/4=(b/2)^2-ac=36-20=16# and
#x_(1,2)=(-b/2+-sqrt(Delta/4))/a=(6+-4)/5rArr#
#x_1=2/5andx_2=2#.
So:
#5x^2-12x+4=5(x-2/5)(x-2)=(5x-2)(x-2)#.
Finally:
#y'=((5x-2)(x-2))/(2sqrt(x-3))#.
