Derivative of (3*tan(2*x)/sin(x)/4)

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

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

      1. Apply the quotient rule, which is:

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

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

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

         1. Rewrite the function to be differentiated:

            $$\tan{\left(2 x \right)} = \frac{\sin{\left(2 x \right)}}{\cos{\left(2 x \right)}}$$

         2. Apply the quotient rule, which is:

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

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

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

            1. Let $u = 2 x$.

            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 $\frac{d}{d x} 2 x$:

               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: $2$

               The result of the chain rule is:

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

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

            1. Let $u = 2 x$.

            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 $\frac{d}{d x} 2 x$:

               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: $2$

               The result of the chain rule is:

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

            Now plug in to the quotient rule:

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

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

         1. The derivative of sine is cosine:

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

         Now plug in to the quotient rule:

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

      So, the result is: $\frac{3 \left(\frac{\left(2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}\right) \sin{\left(x \right)}}{\cos^{2}{\left(2 x \right)}} - \cos{\left(x \right)} \tan{\left(2 x \right)}\right)}{\sin^{2}{\left(x \right)}}$

   So, the result is: $\frac{3 \left(\frac{\left(2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}\right) \sin{\left(x \right)}}{\cos^{2}{\left(2 x \right)}} - \cos{\left(x \right)} \tan{\left(2 x \right)}\right)}{4 \sin^{2}{\left(x \right)}}$

2. Now simplify:

   $$\frac{3 \left(3 \sin{\left(x \right)} + \sin{\left(3 x \right)}\right)}{2 \left(\cos{\left(4 x \right)} + 1\right)}$$

The answer is: