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The derivative of (\csc^{-1}(u)) with respect to (x) is (-\frac{1}{|u|\sqrt{u^2-1}} \cdot \frac{du}{dx}).Here, (u = 5x), so (\frac{du}{dx} = 5).Substituting these values:(\frac{d}{dx} (\csc^{-1}(5x)) = -\frac{1}{|5x|\sqrt{(5x)^2-1}} \cdot 5)(= -\frac{5}{5|x|\sqrt{25x^2-1}})(= -\frac{1}{|x|\sqrt{25x^2-1}}).Assuming (x>0), (|x|=x). Therefore, (= -\frac{1}{x\sqrt{25x^2-1}}).