Derivative of (ctg(1-5x^3))^2

1. Let $u = \cot{\left(1 - 5 x^{3} \right)}$.

2. Apply the power rule: $u^{2}$ goes to $2 u$

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cot{\left(1 - 5 x^{3} \right)}$:

   1. There are multiple ways to do this derivative.

      Method #1

      1. Rewrite the function to be differentiated:

         $$\cot{\left(1 - 5 x^{3} \right)} = - \frac{1}{\tan{\left(5 x^{3} - 1 \right)}}$$

      2. The derivative of a constant times a function is the constant times the derivative of the function.

         1. Let $u = \tan{\left(5 x^{3} - 1 \right)}$.

         2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(5 x^{3} - 1 \right)}$:

            1. Rewrite the function to be differentiated:

               $$\tan{\left(5 x^{3} - 1 \right)} = \frac{\sin{\left(5 x^{3} - 1 \right)}}{\cos{\left(5 x^{3} - 1 \right)}}$$

            2. Apply the quotient rule, which is:

               $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

               $f{\left(x \right)} = \sin{\left(5 x^{3} - 1 \right)}$ and $g{\left(x \right)} = \cos{\left(5 x^{3} - 1 \right)}$.

               To find $\frac{d}{d x} f{\left(x \right)}$:

               1. Let $u = 5 x^{3} - 1$.

               2. The derivative of sine is cosine:

                  $$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$$

               3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x^{3} - 1\right)$:

                  1. Differentiate $5 x^{3} - 1$ term by term:

                     1. The derivative of the constant $-1$ is zero.

                     2. The derivative of a constant times a function is the constant times the derivative of the function.

                        1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$

                        So, the result is: $15 x^{2}$

                     The result is: $15 x^{2}$

                  The result of the chain rule is:

                  $$15 x^{2} \cos{\left(5 x^{3} - 1 \right)}$$

               To find $\frac{d}{d x} g{\left(x \right)}$:

               1. Let $u = 5 x^{3} - 1$.

               2. The derivative of cosine is negative sine:

                  $$\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$$

               3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x^{3} - 1\right)$:

                  1. Differentiate $5 x^{3} - 1$ term by term:

                     1. The derivative of the constant $-1$ is zero.

                     2. The derivative of a constant times a function is the constant times the derivative of the function.

                        1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$

                        So, the result is: $15 x^{2}$

                     The result is: $15 x^{2}$

                  The result of the chain rule is:

                  $$- 15 x^{2} \sin{\left(5 x^{3} - 1 \right)}$$

               Now plug in to the quotient rule:

               $\frac{15 x^{2} \sin^{2}{\left(5 x^{3} - 1 \right)} + 15 x^{2} \cos^{2}{\left(5 x^{3} - 1 \right)}}{\cos^{2}{\left(5 x^{3} - 1 \right)}}$

            The result of the chain rule is:

            $$- \frac{15 x^{2} \sin^{2}{\left(5 x^{3} - 1 \right)} + 15 x^{2} \cos^{2}{\left(5 x^{3} - 1 \right)}}{\cos^{2}{\left(5 x^{3} - 1 \right)} \tan^{2}{\left(5 x^{3} - 1 \right)}}$$

         So, the result is: $\frac{15 x^{2} \sin^{2}{\left(5 x^{3} - 1 \right)} + 15 x^{2} \cos^{2}{\left(5 x^{3} - 1 \right)}}{\cos^{2}{\left(5 x^{3} - 1 \right)} \tan^{2}{\left(5 x^{3} - 1 \right)}}$

      Method #2

      1. Rewrite the function to be differentiated:

         $$\cot{\left(1 - 5 x^{3} \right)} = - \frac{\cos{\left(5 x^{3} - 1 \right)}}{\sin{\left(5 x^{3} - 1 \right)}}$$

      2. The derivative of a constant times a function is the constant times the derivative of the function.

         1. Apply the quotient rule, which is:

            $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

            $f{\left(x \right)} = \cos{\left(5 x^{3} - 1 \right)}$ and $g{\left(x \right)} = \sin{\left(5 x^{3} - 1 \right)}$.

            To find $\frac{d}{d x} f{\left(x \right)}$:

            1. Let $u = 5 x^{3} - 1$.

            2. The derivative of cosine is negative sine:

               $$\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$$

            3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x^{3} - 1\right)$:

               1. Differentiate $5 x^{3} - 1$ term by term:

                  1. The derivative of the constant $-1$ is zero.

                  2. The derivative of a constant times a function is the constant times the derivative of the function.

                     1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$

                     So, the result is: $15 x^{2}$

                  The result is: $15 x^{2}$

               The result of the chain rule is:

               $$- 15 x^{2} \sin{\left(5 x^{3} - 1 \right)}$$

            To find $\frac{d}{d x} g{\left(x \right)}$:

            1. Let $u = 5 x^{3} - 1$.

            2. The derivative of sine is cosine:

               $$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$$

            3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x^{3} - 1\right)$:

               1. Differentiate $5 x^{3} - 1$ term by term:

                  1. The derivative of the constant $-1$ is zero.

                  2. The derivative of a constant times a function is the constant times the derivative of the function.

                     1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$

                     So, the result is: $15 x^{2}$

                  The result is: $15 x^{2}$

               The result of the chain rule is:

               $$15 x^{2} \cos{\left(5 x^{3} - 1 \right)}$$

            Now plug in to the quotient rule:

            $\frac{- 15 x^{2} \sin^{2}{\left(5 x^{3} - 1 \right)} - 15 x^{2} \cos^{2}{\left(5 x^{3} - 1 \right)}}{\sin^{2}{\left(5 x^{3} - 1 \right)}}$

         So, the result is: $- \frac{- 15 x^{2} \sin^{2}{\left(5 x^{3} - 1 \right)} - 15 x^{2} \cos^{2}{\left(5 x^{3} - 1 \right)}}{\sin^{2}{\left(5 x^{3} - 1 \right)}}$

   The result of the chain rule is:

   $$\frac{2 \left(15 x^{2} \sin^{2}{\left(5 x^{3} - 1 \right)} + 15 x^{2} \cos^{2}{\left(5 x^{3} - 1 \right)}\right) \cot{\left(1 - 5 x^{3} \right)}}{\cos^{2}{\left(5 x^{3} - 1 \right)} \tan^{2}{\left(5 x^{3} - 1 \right)}}$$

4. Now simplify:

   $$- \frac{30 x^{2} \cos{\left(5 x^{3} - 1 \right)}}{\sin^{3}{\left(5 x^{3} - 1 \right)}}$$

The answer is:

$$- \frac{30 x^{2} \cos{\left(5 x^{3} - 1 \right)}}{\sin^{3}{\left(5 x^{3} - 1 \right)}}$$