Derivative of sec5x

1. Rewrite the function to be differentiated:

   $$\sec{\left(5 x \right)} = \frac{1}{\cos{\left(5 x \right)}}$$

2. Let $u = \cos{\left(5 x \right)}$.

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

4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(5 x \right)}$:

   1. Let $u = 5 x$.

   2. The derivative of cosine is negative sine:

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$:

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

         1. Apply the power rule: $x$ goes to $1$

         So, the result is: $5$

      The result of the chain rule is:

      $$- 5 \sin{\left(5 x \right)}$$

   The result of the chain rule is:

   $$\frac{5 \sin{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}}$$

The answer is: