Derivative of (sec(x))^2-1

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

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

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

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

      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:

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

   4. The derivative of the constant $\left(-1\right) 1$ is zero.

   The result is: $\frac{2 \sin{\left(x \right)} \sec{\left(x \right)}}{\cos^{2}{\left(x \right)}}$

2. Now simplify:

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

The answer is: