Derivative of y=sin(8x)+2^x

1. Differentiate $2^{x} + \sin{\left(8 x \right)}$ term by term:

   1. Let $u = 8 x$.

   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 $\frac{d}{d x} 8 x$:

      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: $8$

      The result of the chain rule is:

      $$8 \cos{\left(8 x \right)}$$

   4. $\frac{d}{d x} 2^{x} = 2^{x} \log{\left(2 \right)}$

   The result is: $2^{x} \log{\left(2 \right)} + 8 \cos{\left(8 x \right)}$

The answer is:

$$2^{x} \log{\left(2 \right)} + 8 \cos{\left(8 x \right)}$$