Derivative of sin(x)-x*(cos(x))^2

1. Differentiate $- x \cos^{2}{\left(x \right)} + \sin{\left(x \right)}$ term by term:

   1. The derivative of sine is cosine:

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

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

      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 $1$

         $g{\left(x \right)} = \cos^{2}{\left(x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

         1. Let $u = \cos{\left(x \right)}$.

         2. Apply the power rule: $u^{2}$ goes to $2 u$

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

            1. The derivative of cosine is negative sine:

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

            The result of the chain rule is:

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

         The result is: $- 2 x \sin{\left(x \right)} \cos{\left(x \right)} + \cos^{2}{\left(x \right)}$

      So, the result is: $2 x \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}$

   The result is: $2 x \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)} + \cos{\left(x \right)}$

2. Now simplify:

   $$\left(2 x \sin{\left(x \right)} - \cos{\left(x \right)} + 1\right) \cos{\left(x \right)}$$

The answer is:

$$\left(2 x \sin{\left(x \right)} - \cos{\left(x \right)} + 1\right) \cos{\left(x \right)}$$