Derivative of sec^3x

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

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

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:

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

4. Now simplify:

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

The answer is: