Answer
$$ y'= -35x^4\cot^6(x^5)\csc^2( x^5).$$
Work Step by Step
Since $ y=\cot^7x^5$, by using the chain rule: $(f(g(x)))^{\prime}=f^{\prime}(g(x)) g^{\prime}(x)$ and recalling that $(\cot x)'=-\csc^2 x $, the derivative $ y'$ is given by
$$ y'=7\cot^6x^5 (\cot x^5)'=7\cot^6x^5 (-\csc^2 x^5)( x^5)'
\\=7\cot^6x^5 (-\csc^2 x^5)(5x^4)=-35x^4\cot^6(x^5)\csc^2( x^5).$$
