Derivative of 1/cos^2(5x-1)

1. Apply the quotient rule, which is:

   $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

   $f{\left(x \right)} = 1$ and $g{\left(x \right)} = \cos^{2}{\left(5 x - 1 \right)}$.

   To find $\frac{d}{d x} f{\left(x \right)}$:

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

   To find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = \cos{\left(5 x - 1 \right)}$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(5 x - 1 \right)}$:

      1. Let $u = 5 x - 1$.

      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 $\frac{d}{d x} \left(5 x - 1\right)$:

         1. Differentiate $5 x - 1$ term by term:

            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: $5$

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

            The result is: $5$

         The result of the chain rule is:

         $$- 5 \sin{\left(5 x - 1 \right)}$$

      The result of the chain rule is:

      $$- 10 \sin{\left(5 x - 1 \right)} \cos{\left(5 x - 1 \right)}$$

   Now plug in to the quotient rule:

   $\frac{10 \sin{\left(5 x - 1 \right)}}{\cos^{3}{\left(5 x - 1 \right)}}$

2. Now simplify:

   $$\frac{10 \sin{\left(5 x - 1 \right)}}{\cos^{3}{\left(5 x - 1 \right)}}$$

The answer is: