Derivative of sin(6x)cos(5x)tg(7x)

1. Apply the product rule:

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

   $f{\left(x \right)} = \sin{\left(6 x \right)} \cos{\left(5 x \right)}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Apply the product rule:

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

      $f{\left(x \right)} = \sin{\left(6 x \right)}$; to find $\frac{d}{d x} f{\left(x \right)}$:

      1. Let $u = 6 x$.

      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 $\frac{d}{d x} 6 x$:

         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: $6$

         The result of the chain rule is:

         $$6 \cos{\left(6 x \right)}$$

      $g{\left(x \right)} = \cos{\left(5 x \right)}$; 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:

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

      The result is: $- 5 \sin{\left(5 x \right)} \sin{\left(6 x \right)} + 6 \cos{\left(5 x \right)} \cos{\left(6 x \right)}$

   $g{\left(x \right)} = \tan{\left(7 x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Rewrite the function to be differentiated:

      $$\tan{\left(7 x \right)} = \frac{\sin{\left(7 x \right)}}{\cos{\left(7 x \right)}}$$

   2. Apply the quotient rule, which is:

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

      $f{\left(x \right)} = \sin{\left(7 x \right)}$ and $g{\left(x \right)} = \cos{\left(7 x \right)}$.

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

      1. Let $u = 7 x$.

      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 $\frac{d}{d x} 7 x$:

         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: $7$

         The result of the chain rule is:

         $$7 \cos{\left(7 x \right)}$$

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

      1. Let $u = 7 x$.

      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 $\frac{d}{d x} 7 x$:

         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: $7$

         The result of the chain rule is:

         $$- 7 \sin{\left(7 x \right)}$$

      Now plug in to the quotient rule:

      $\frac{7 \sin^{2}{\left(7 x \right)} + 7 \cos^{2}{\left(7 x \right)}}{\cos^{2}{\left(7 x \right)}}$

   The result is: $\left(- 5 \sin{\left(5 x \right)} \sin{\left(6 x \right)} + 6 \cos{\left(5 x \right)} \cos{\left(6 x \right)}\right) \tan{\left(7 x \right)} + \frac{\left(7 \sin^{2}{\left(7 x \right)} + 7 \cos^{2}{\left(7 x \right)}\right) \sin{\left(6 x \right)} \cos{\left(5 x \right)}}{\cos^{2}{\left(7 x \right)}}$

2. Now simplify:

   $$\frac{\frac{\left(\cos{\left(x \right)} + 11 \cos{\left(11 x \right)}\right) \sin{\left(7 x \right)} \cos{\left(7 x \right)}}{2} + 7 \sin{\left(6 x \right)} \cos{\left(5 x \right)}}{\cos^{2}{\left(7 x \right)}}$$

The answer is:

$$\frac{\frac{\left(\cos{\left(x \right)} + 11 \cos{\left(11 x \right)}\right) \sin{\left(7 x \right)} \cos{\left(7 x \right)}}{2} + 7 \sin{\left(6 x \right)} \cos{\left(5 x \right)}}{\cos^{2}{\left(7 x \right)}}$$