Given a function #f# continuous on an interval #[a,b]# (with #a < b#) and a number #c\in [a,b]#, the Fundamental Theorem of Calculus ("Part 1"...actually, some books call it "Part 2") says that #d/dx(\int_{c}^{x}f(v)\ dv)=f(x)# for all #x\in [a,b]#. Thought of another way, the function #F# defined by the equation #F(x)=\int_{c}^{x}f(v)\ dv# is an antiderivative of #f# on the interval #[a,b]# (also, #F(c)=0#).
For the problem at hand, let #F(x)=\int_{1}^{x}(6+v^8)^3\ dv# and let #g(x)=cos(x)# so that we must find #d/dx(F(g(x))#. This also requires the Chain Rule, which gives #d/dx(F(g(x)))=F'(g(x))*g'(x)#. But #F'(x)=(6+x^8)^3# and #g'(x)=-sin(x)#. The final answer is therefore:
#d/dx(\int_{1}^{\cos(x)}(6+v^8)^3\ dv)=-(6+cos^8(x))^3*sin(x)#
