Derivative of sec⁴(x)-2tan²x

1. Differentiate $- 2 \tan^{2}{\left(x \right)} + \sec^{4}{\left(x \right)}$ term by term:

   1. Let $u = \sec{\left(x \right)}$.

   2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sec{\left(x \right)}$:

      1. Rewrite the function to be differentiated:

         $$\sec{\left(x \right)} = \frac{1}{\cos{\left(x \right)}}$$

      2. Let $u = \cos{\left(x \right)}$.

      3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$

      4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$:

         1. The derivative of cosine is negative sine:

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

         The result of the chain rule is:

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

      The result of the chain rule is:

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

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

      1. Let $u = \tan{\left(x \right)}$.

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$:

         1. Rewrite the function to be differentiated:

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

         2. Apply the quotient rule, which is:

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

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

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

            1. The derivative of sine is cosine:

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

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

            1. The derivative of cosine is negative sine:

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

            Now plug in to the quotient rule:

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

         The result of the chain rule is:

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

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

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

2. Now simplify:

   $$\frac{4 \sin^{3}{\left(x \right)}}{\cos^{5}{\left(x \right)}}$$

The answer is:

$$\frac{4 \sin^{3}{\left(x \right)}}{\cos^{5}{\left(x \right)}}$$