Derivative of 1-sec^2x

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

   1. The derivative of the constant $1$ is zero.

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

      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:

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

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

   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:

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