Derivative of y=sin(3x+1)+cos5x

1. Differentiate $\sin{\left(3 x + 1 \right)} + \cos{\left(5 x \right)}$ term by term:

   1. Let $u = 3 x + 1$.

   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 $\frac{d}{d x} \left(3 x + 1\right)$:

      1. Differentiate $3 x + 1$ 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: $3$

         2. The derivative of the constant $1$ is zero.

         The result is: $3$

      The result of the chain rule is:

      $$3 \cos{\left(3 x + 1 \right)}$$

   4. Let $u = 5 x$.

   5. The derivative of cosine is negative sine:

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

   6. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$:

      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: $5$

      The result of the chain rule is:

      $$- 5 \sin{\left(5 x \right)}$$

   The result is: $- 5 \sin{\left(5 x \right)} + 3 \cos{\left(3 x + 1 \right)}$

2. Now simplify:

   $$- 5 \sin{\left(5 x \right)} + 3 \cos{\left(3 x + 1 \right)}$$

The answer is:

$$- 5 \sin{\left(5 x \right)} + 3 \cos{\left(3 x + 1 \right)}$$