Derivative of x^2*(tan(4x+1))

1. Apply the product rule:

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

   $f{\left(x \right)} = x^{2}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Apply the power rule: $x^{2}$ goes to $2 x$

   $g{\left(x \right)} = \tan{\left(4 x + 1 \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Rewrite the function to be differentiated:

      $$\tan{\left(4 x + 1 \right)} = \frac{\sin{\left(4 x + 1 \right)}}{\cos{\left(4 x + 1 \right)}}$$

   2. Apply the quotient rule, which is:

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

      $f{\left(x \right)} = \sin{\left(4 x + 1 \right)}$ and $g{\left(x \right)} = \cos{\left(4 x + 1 \right)}$.

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

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

      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 $\frac{d}{d x} \left(4 x + 1\right)$:

         1. Differentiate $4 x + 1$ 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 $1$ is zero.

            The result is: $4$

         The result of the chain rule is:

         $$4 \cos{\left(4 x + 1 \right)}$$

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

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

      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 $\frac{d}{d x} \left(4 x + 1\right)$:

         1. Differentiate $4 x + 1$ 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 $1$ is zero.

            The result is: $4$

         The result of the chain rule is:

         $$- 4 \sin{\left(4 x + 1 \right)}$$

      Now plug in to the quotient rule:

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

   The result is: $\frac{x^{2} \left(4 \sin^{2}{\left(4 x + 1 \right)} + 4 \cos^{2}{\left(4 x + 1 \right)}\right)}{\cos^{2}{\left(4 x + 1 \right)}} + 2 x \tan{\left(4 x + 1 \right)}$

2. Now simplify:

   $$\frac{4 x^{2}}{\cos^{2}{\left(4 x + 1 \right)}} + 2 x \tan{\left(4 x + 1 \right)}$$

The answer is:

$$\frac{4 x^{2}}{\cos^{2}{\left(4 x + 1 \right)}} + 2 x \tan{\left(4 x + 1 \right)}$$