Derivative of y=tg(8x²+3x+4)

1. Rewrite the function to be differentiated:

   $$\tan{\left(\left(8 x^{2} + 3 x\right) + 4 \right)} = \frac{\sin{\left(\left(8 x^{2} + 3 x\right) + 4 \right)}}{\cos{\left(\left(8 x^{2} + 3 x\right) + 4 \right)}}$$

2. Apply the quotient rule, which is:

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

   $f{\left(x \right)} = \sin{\left(\left(8 x^{2} + 3 x\right) + 4 \right)}$ and $g{\left(x \right)} = \cos{\left(\left(8 x^{2} + 3 x\right) + 4 \right)}$.

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

   1. Let $u = \left(8 x^{2} + 3 x\right) + 4$.

   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 $\frac{d}{d x} \left(\left(8 x^{2} + 3 x\right) + 4\right)$:

      1. Differentiate $\left(8 x^{2} + 3 x\right) + 4$ term by term:

         1. Differentiate $8 x^{2} + 3 x$ 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: $x^{2}$ goes to $2 x$

               So, the result is: $16 x$

            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: $16 x + 3$

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

         The result is: $16 x + 3$

      The result of the chain rule is:

      $$\left(16 x + 3\right) \cos{\left(\left(8 x^{2} + 3 x\right) + 4 \right)}$$

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

   1. Let $u = \left(8 x^{2} + 3 x\right) + 4$.

   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 $\frac{d}{d x} \left(\left(8 x^{2} + 3 x\right) + 4\right)$:

      1. Differentiate $\left(8 x^{2} + 3 x\right) + 4$ term by term:

         1. Differentiate $8 x^{2} + 3 x$ 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: $x^{2}$ goes to $2 x$

               So, the result is: $16 x$

            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: $16 x + 3$

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

         The result is: $16 x + 3$

      The result of the chain rule is:

      $$- \left(16 x + 3\right) \sin{\left(\left(8 x^{2} + 3 x\right) + 4 \right)}$$

   Now plug in to the quotient rule:

   $\frac{\left(16 x + 3\right) \sin^{2}{\left(\left(8 x^{2} + 3 x\right) + 4 \right)} + \left(16 x + 3\right) \cos^{2}{\left(\left(8 x^{2} + 3 x\right) + 4 \right)}}{\cos^{2}{\left(\left(8 x^{2} + 3 x\right) + 4 \right)}}$

3. Now simplify:

   $$\frac{16 x + 3}{\cos^{2}{\left(8 x^{2} + 3 x + 4 \right)}}$$

The answer is:

$$\frac{16 x + 3}{\cos^{2}{\left(8 x^{2} + 3 x + 4 \right)}}$$