Derivative of a*sin(10*x)+b*cos(10*x)+c*e^(10*x)

1. Differentiate $e^{10 x} c + \left(a \sin{\left(10 x \right)} + b \cos{\left(10 x \right)}\right)$ term by term:

   1. Differentiate $a \sin{\left(10 x \right)} + b \cos{\left(10 x \right)}$ term by term:

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

         1. Let $u = 10 x$.

         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:

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

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

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

         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:

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

         So, the result is: $- 10 b \sin{\left(10 x \right)}$

      The result is: $10 a \cos{\left(10 x \right)} - 10 b \sin{\left(10 x \right)}$

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

      1. Let $u = 10 x$.

      2. The derivative of $e^{u}$ is itself.

      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:

         $$10 e^{10 x}$$

      So, the result is: $10 c e^{10 x}$

   The result is: $10 a \cos{\left(10 x \right)} - 10 b \sin{\left(10 x \right)} + 10 c e^{10 x}$

The answer is:

$$10 a \cos{\left(10 x \right)} - 10 b \sin{\left(10 x \right)} + 10 c e^{10 x}$$