Chapter 2 - The Derivative - 2.6 The Chain Rule - Exercises Set 2.6 - Page 158: 25

$$f'(x)=-3(x+\csc(x^3+3))^{-4}(1-3x^2\csc(x^3+3)\cot(x^3+3))$$

$$f'(x)=\Big((x+\csc(x^3+3))^{-3}\Big)'=-3(x+\csc(x^3+3))^{-4}\cdot(x+\csc(x^3+3))'= -3(x+\csc(x^3+3))^{-4}(1+(-\csc(x^3-3)\cot(x^3-3))\cdot(x^3+3)')= -3(x+\csc(x^3+3))^{-4}(1-\csc(x^3+3)\cot(x^3+3)\cdot3x^2)= -3(x+\csc(x^3+3))^{-4}(1-3x^2\csc(x^3+3)\cot(x^3+3))$$