Derivative of с^(sin(a*x+b))

1. Let $u = \sin{\left(a x + b \right)}$.

2. $\frac{\partial}{\partial u} c^{u} = c^{u} \log{\left(c \right)}$

3. Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} \sin{\left(a x + b \right)}$:

   1. Let $u = a x + b$.

   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 $\frac{\partial}{\partial x} \left(a x + b\right)$:

      1. Differentiate $a x + b$ 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: $a$

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

         The result is: $a$

      The result of the chain rule is:

      $$a \cos{\left(a x + b \right)}$$

   The result of the chain rule is:

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

4. Now simplify:

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

The answer is: