Derivative of v*cos(ax)+u*sin(ax)

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

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

      1. Let $u = a x$.

      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 $\frac{\partial}{\partial x} a x$:

         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: $a$

         The result of the chain rule is:

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

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

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

      1. Let $u = a x$.

      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 $\frac{\partial}{\partial x} a x$:

         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: $a$

         The result of the chain rule is:

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

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

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

2. Now simplify:

   $$a \left(u \cos{\left(a x \right)} - v \sin{\left(a x \right)}\right)$$

The answer is:

$$a \left(u \cos{\left(a x \right)} - v \sin{\left(a x \right)}\right)$$