Derivative of tan(3*x-4)*cos(x)

1. Apply the product rule:

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

   $f{\left(x \right)} = \tan{\left(3 x - 4 \right)}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Rewrite the function to be differentiated:

      $$\tan{\left(3 x - 4 \right)} = \frac{\sin{\left(3 x - 4 \right)}}{\cos{\left(3 x - 4 \right)}}$$

   2. Apply the quotient rule, which is:

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

      $f{\left(x \right)} = \sin{\left(3 x - 4 \right)}$ and $g{\left(x \right)} = \cos{\left(3 x - 4 \right)}$.

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

      1. Let $u = 3 x - 4$.

      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 $\frac{d}{d x} \left(3 x - 4\right)$:

         1. Differentiate $3 x - 4$ term by term:

            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: $3$

            2. The derivative of the constant $-4$ is zero.

            The result is: $3$

         The result of the chain rule is:

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

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

      1. Let $u = 3 x - 4$.

      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 $\frac{d}{d x} \left(3 x - 4\right)$:

         1. Differentiate $3 x - 4$ term by term:

            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: $3$

            2. The derivative of the constant $-4$ is zero.

            The result is: $3$

         The result of the chain rule is:

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

      Now plug in to the quotient rule:

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

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

   1. The derivative of cosine is negative sine:

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

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

2. Now simplify:

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

The answer is:

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