Derivative of y=ln^3(sec(x))

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

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

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

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

   2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$.

   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:

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

   The result of the chain rule is:

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

4. Now simplify:

   $$3 \log{\left(\frac{1}{\cos{\left(x \right)}} \right)}^{2} \tan{\left(x \right)}$$

The answer is: