Derivative of tan(5*x)*ctg(10x)

1. Apply the product rule:

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

   $f{\left(x \right)} = \tan{\left(5 x \right)}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   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:

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

   $g{\left(x \right)} = \cot{\left(10 x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. There are multiple ways to do this derivative.

      Method #1

      1. Rewrite the function to be differentiated:

         $$\cot{\left(10 x \right)} = \frac{1}{\tan{\left(10 x \right)}}$$

      2. Let $u = \tan{\left(10 x \right)}$.

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

      4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(10 x \right)}$:

         1. Let $u = 10 x$.

         2. $$\frac{d}{d u} \tan{\left(u \right)} = \frac{1}{\cos^{2}{\left(u \right)}}$$

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 10 x$:

            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: $10$

            The result of the chain rule is:

            $$\frac{10}{\cos^{2}{\left(10 x \right)}}$$

         The result of the chain rule is:

         $$- \frac{10 \sin^{2}{\left(10 x \right)} + 10 \cos^{2}{\left(10 x \right)}}{\cos^{2}{\left(10 x \right)} \tan^{2}{\left(10 x \right)}}$$

      Method #2

      1. Rewrite the function to be differentiated:

         $$\cot{\left(10 x \right)} = \frac{\cos{\left(10 x \right)}}{\sin{\left(10 x \right)}}$$

      2. Apply the quotient rule, which is:

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

         $f{\left(x \right)} = \cos{\left(10 x \right)}$ and $g{\left(x \right)} = \sin{\left(10 x \right)}$.

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

         1. Let $u = 10 x$.

         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 $\frac{d}{d x} 10 x$:

            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: $10$

            The result of the chain rule is:

            $$- 10 \sin{\left(10 x \right)}$$

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

         1. Let $u = 10 x$.

         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 $\frac{d}{d x} 10 x$:

            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: $10$

            The result of the chain rule is:

            $$10 \cos{\left(10 x \right)}$$

         Now plug in to the quotient rule:

         $\frac{- 10 \sin^{2}{\left(10 x \right)} - 10 \cos^{2}{\left(10 x \right)}}{\sin^{2}{\left(10 x \right)}}$

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

2. Now simplify:

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

The answer is:

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