Derivative of 9^(sin(7*x-2))

1. Let $u = \sin{\left(7 x - 2 \right)}$.

2. $\frac{d}{d u} 9^{u} = 9^{u} \log{\left(9 \right)}$

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(7 x - 2 \right)}$:

   1. Let $u = 7 x - 2$.

   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 $\frac{d}{d x} \left(7 x - 2\right)$:

      1. Differentiate $7 x - 2$ 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: $7$

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

         The result is: $7$

      The result of the chain rule is:

      $$7 \cos{\left(7 x - 2 \right)}$$

   The result of the chain rule is:

   $$7 \cdot 9^{\sin{\left(7 x - 2 \right)}} \log{\left(9 \right)} \cos{\left(7 x - 2 \right)}$$

4. Now simplify:

   $$14 \cdot 9^{\sin{\left(7 x - 2 \right)}} \log{\left(3 \right)} \cos{\left(7 x - 2 \right)}$$

The answer is: