Use the power chain rule [tex]\frac{d}{dx}f(x)^n=nf(x)^n^-^1[/tex] × [tex]\frac{d}{dx}f(x)[/tex]
For this problem, [tex]f(x)=x^5+7x[/tex]. Find its derivative to get [tex]\frac{d}{dx} f(x)=5x^4+7[/tex].
[tex]n=8[/tex], so [tex]n-1=7[/tex].
[tex]\frac{d}{dx}f(x)^n=nf(x)^n^-^1[/tex] × [tex]\frac{d}{dx}f(x)[/tex]
→ [tex]\frac{d}{dx}(x^5+7x)^8=8(x^5+7x)^7[/tex] × [tex](5x^4+7)[/tex]
For clarity, × is a multiplication sign. Let's make the final answer look a little nicer → [tex]8(5x^4+7)(x^5+7x)^7[/tex]