Derivative of ((2x)/(x+2))-3sinx

1. Differentiate $\frac{2 x}{x + 2} - 3 \sin{\left(x \right)}$ term by term:

   1. Apply the quotient rule, which is:

      $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

      $f{\left(x \right)} = 2 x$ and $g{\left(x \right)} = x + 2$.

      To find $\frac{d}{d x} f{\left(x \right)}$:

      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: $2$

      To find $\frac{d}{d x} g{\left(x \right)}$:

      1. Differentiate $x + 2$ term by term:

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

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

         The result is: $1$

      Now plug in to the quotient rule:

      $\frac{4}{\left(x + 2\right)^{2}}$

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

      1. The derivative of sine is cosine:

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

      So, the result is: $- 3 \cos{\left(x \right)}$

   The result is: $- 3 \cos{\left(x \right)} + \frac{4}{\left(x + 2\right)^{2}}$

The answer is: