Derivative of −4e^3x+6sin(2x)+7cos(−7x)−3

1. Differentiate $\left(\left(- 4 e^{3} x + 6 \sin{\left(2 x \right)}\right) + 7 \cos{\left(- 7 x \right)}\right) - 3$ term by term:

   1. Differentiate $\left(- 4 e^{3} x + 6 \sin{\left(2 x \right)}\right) + 7 \cos{\left(- 7 x \right)}$ term by term:

      1. Differentiate $- 4 e^{3} x + 6 \sin{\left(2 x \right)}$ term by term:

         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: $- 4 e^{3}$

         2. 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: $12 \cos{\left(2 x \right)}$

         The result is: $12 \cos{\left(2 x \right)} - 4 e^{3}$

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

         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 $\frac{d}{d x} \left(- 7 x\right)$:

            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:

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

         So, the result is: $- 49 \sin{\left(7 x \right)}$

      The result is: $- 49 \sin{\left(7 x \right)} + 12 \cos{\left(2 x \right)} - 4 e^{3}$

   2. The derivative of the constant $-3$ is zero.

   The result is: $- 49 \sin{\left(7 x \right)} + 12 \cos{\left(2 x \right)} - 4 e^{3}$

The answer is:

$$- 49 \sin{\left(7 x \right)} + 12 \cos{\left(2 x \right)} - 4 e^{3}$$