Derivative of (sin3x)/cos(3x)^5

1. Apply the quotient rule, which is:

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

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

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

   1. Let $u = 3 x$.

   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 $\frac{d}{d x} 3 x$:

      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: $3$

      The result of the chain rule is:

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

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

   1. Let $u = \cos{\left(3 x \right)}$.

   2. Apply the power rule: $u^{5}$ goes to $5 u^{4}$

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

      1. Let $u = 3 x$.

      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 $\frac{d}{d x} 3 x$:

         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: $3$

         The result of the chain rule is:

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

      The result of the chain rule is:

      $$- 15 \sin{\left(3 x \right)} \cos^{4}{\left(3 x \right)}$$

   Now plug in to the quotient rule:

   $\frac{15 \sin^{2}{\left(3 x \right)} \cos^{4}{\left(3 x \right)} + 3 \cos^{6}{\left(3 x \right)}}{\cos^{10}{\left(3 x \right)}}$

2. Now simplify:

   $$\frac{3 \cdot \left(3 - 2 \cos{\left(6 x \right)}\right)}{\cos^{6}{\left(3 x \right)}}$$

The answer is: