Derivative of y(x)=5sin2x-2ctg5x

1. Differentiate $5 \sin{\left(2 x \right)} - 2 \cot{\left(5 x \right)}$ term by term:

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

      1. Let $u = 2 x$.

      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 $\frac{d}{d x} 2 x$:

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

            1. Apply the power rule: $x$ goes to $1$

            So, the result is: $2$

         The result of the chain rule is:

         $$2 \cos{\left(2 x \right)}$$

      So, the result is: $10 \cos{\left(2 x \right)}$

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

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

         1. There are multiple ways to do this derivative.

            Method #1

            1. Rewrite the function to be differentiated:

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

            2. Let $u = \tan{\left(5 x \right)}$.

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

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

               1. Rewrite the function to be differentiated:

                  $$\tan{\left(5 x \right)} = \frac{\sin{\left(5 x \right)}}{\cos{\left(5 x \right)}}$$

               2. Apply the quotient rule, which is:

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

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

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

                  1. Let $u = 5 x$.

                  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 $\frac{d}{d x} 5 x$:

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

                        1. Apply the power rule: $x$ goes to $1$

                        So, the result is: $5$

                     The result of the chain rule is:

                     $$5 \cos{\left(5 x \right)}$$

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

                  1. Let $u = 5 x$.

                  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 $\frac{d}{d x} 5 x$:

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

                        1. Apply the power rule: $x$ goes to $1$

                        So, the result is: $5$

                     The result of the chain rule is:

                     $$- 5 \sin{\left(5 x \right)}$$

                  Now plug in to the quotient rule:

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

               The result of the chain rule is:

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

            Method #2

            1. Rewrite the function to be differentiated:

               $$\cot{\left(5 x \right)} = \frac{\cos{\left(5 x \right)}}{\sin{\left(5 x \right)}}$$

            2. Apply the quotient rule, which is:

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

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

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

               1. Let $u = 5 x$.

               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 $\frac{d}{d x} 5 x$:

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

                     1. Apply the power rule: $x$ goes to $1$

                     So, the result is: $5$

                  The result of the chain rule is:

                  $$- 5 \sin{\left(5 x \right)}$$

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

               1. Let $u = 5 x$.

               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 $\frac{d}{d x} 5 x$:

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

                     1. Apply the power rule: $x$ goes to $1$

                     So, the result is: $5$

                  The result of the chain rule is:

                  $$5 \cos{\left(5 x \right)}$$

               Now plug in to the quotient rule:

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

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

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

   The result is: $\frac{2 \cdot \left(5 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}\right)}{\cos^{2}{\left(5 x \right)} \tan^{2}{\left(5 x \right)}} + 10 \cos{\left(2 x \right)}$

2. Now simplify:

   $$10 \cos{\left(2 x \right)} + \frac{20}{1 - \cos{\left(10 x \right)}}$$

The answer is: