Derivative of y=cos(x)/(3*x+5)

1. Apply the quotient rule, which is:

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

   $f{\left(x \right)} = \cos{\left(x \right)}$ and $g{\left(x \right)} = 3 x + 5$.

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

   1. The derivative of cosine is negative sine:

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

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

   1. Differentiate $3 x + 5$ term by term:

      1. The derivative of the constant $5$ is zero.

      2. 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: $3$

      The result is: $3$

   Now plug in to the quotient rule:

   $\frac{- \left(3 x + 5\right) \sin{\left(x \right)} - 3 \cos{\left(x \right)}}{\left(3 x + 5\right)^{2}}$

2. Now simplify:

   $$- \frac{\left(3 x + 5\right) \sin{\left(x \right)} + 3 \cos{\left(x \right)}}{\left(3 x + 5\right)^{2}}$$

The answer is:

$$- \frac{\left(3 x + 5\right) \sin{\left(x \right)} + 3 \cos{\left(x \right)}}{\left(3 x + 5\right)^{2}}$$