Derivative of sin(x)+(3*x)^6

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

   1. The derivative of sine is cosine:

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

   2. Let $u = 3 x$.

   3. Apply the power rule: $u^{6}$ goes to $6 u^{5}$

   4. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$:

      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: $3$

      The result of the chain rule is:

      $$4374 x^{5}$$

   The result is: $4374 x^{5} + \cos{\left(x \right)}$

The answer is:

$$4374 x^{5} + \cos{\left(x \right)}$$