Derivative of tg(4x+5)

1. Rewrite the function to be differentiated:

   $$\tan{\left(4 x + 5 \right)} = \frac{\sin{\left(4 x + 5 \right)}}{\cos{\left(4 x + 5 \right)}}$$

2. Apply the quotient rule, which is:

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

   $f{\left(x \right)} = \sin{\left(4 x + 5 \right)}$ and $g{\left(x \right)} = \cos{\left(4 x + 5 \right)}$.

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

   1. Let $u = 4 x + 5$.

   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 $\frac{d}{d x} \left(4 x + 5\right)$:

      1. Differentiate $4 x + 5$ 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: $4$

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

         The result is: $4$

      The result of the chain rule is:

      $$4 \cos{\left(4 x + 5 \right)}$$

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

   1. Let $u = 4 x + 5$.

   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 $\frac{d}{d x} \left(4 x + 5\right)$:

      1. Differentiate $4 x + 5$ 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: $4$

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

         The result is: $4$

      The result of the chain rule is:

      $$- 4 \sin{\left(4 x + 5 \right)}$$

   Now plug in to the quotient rule:

   $\frac{4 \sin^{2}{\left(4 x + 5 \right)} + 4 \cos^{2}{\left(4 x + 5 \right)}}{\cos^{2}{\left(4 x + 5 \right)}}$

3. Now simplify:

   $$\frac{4}{\cos^{2}{\left(4 x + 5 \right)}}$$

The answer is:

$$\frac{4}{\cos^{2}{\left(4 x + 5 \right)}}$$