#y=e^8sqrt(x+x^6)#
First of all, note that #e^8# is a (multiplicative) constant. It will remain on the outside of the function as we differentiate it.
For the #sqrt(x+x^6)# portion, we should rewrite it as #(x+x^6)^(1/2)# and then differentiate it using the product and chain rules.
Since the derivative of #x^(1/2)# is #1/2x^(-1/2)#, the derivative of #f(x)^(1/2)# is #1/2f(x)^(-1/2)*f'(x)#.
Thus, the derivative of the function is:
#dy/dx=e^8(1/2(x+x^6)^(-1/2))*d/dx(x+x^6)#
The derivative of #x+x^6# is #1+6x^5#:
#dy/dx=(e^8(1+6x^5))/(2sqrt(x+x^6))#
