Derivative of x*cos(x+3)+7

1. Differentiate $x \cos{\left(x + 3 \right)} + 7$ term by term:

   1. Apply the product rule:

      $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

      $f{\left(x \right)} = x$; to find $\frac{d}{d x} f{\left(x \right)}$:

      1. Apply the power rule: $x$ goes to $1$

      $g{\left(x \right)} = \cos{\left(x + 3 \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

      1. Let $u = x + 3$.

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

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

            1. Apply the power rule: $x$ goes to $1$

            2. The derivative of the constant $3$ is zero.

            The result is: $1$

         The result of the chain rule is:

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

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

   2. The derivative of the constant $7$ is zero.

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

2. Now simplify:

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

The answer is:

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