Derivative of sin(7*x)*cos(x)-sin(x)*cos(7*x)

1. Differentiate $- \sin{\left(x \right)} \cos{\left(7 x \right)} + \sin{\left(7 x \right)} \cos{\left(x \right)}$ term by term:

   1. Apply the product rule:

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

      $f{\left(x \right)} = \sin{\left(7 x \right)}$; 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:

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

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

      1. The derivative of cosine is negative sine:

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

      The result is: $- \sin{\left(x \right)} \sin{\left(7 x \right)} + 7 \cos{\left(x \right)} \cos{\left(7 x \right)}$

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

      1. Apply the product rule:

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

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

         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:

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

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

         1. The derivative of sine is cosine:

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

         The result is: $- 7 \sin{\left(x \right)} \sin{\left(7 x \right)} + \cos{\left(x \right)} \cos{\left(7 x \right)}$

      So, the result is: $7 \sin{\left(x \right)} \sin{\left(7 x \right)} - \cos{\left(x \right)} \cos{\left(7 x \right)}$

   The result is: $6 \sin{\left(x \right)} \sin{\left(7 x \right)} + 6 \cos{\left(x \right)} \cos{\left(7 x \right)}$

2. Now simplify:

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

The answer is:

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