Derivative of y=sec(2x²-1)

1. Rewrite the function to be differentiated:

   $$\sec{\left(2 x^{2} - 1 \right)} = \frac{1}{\cos{\left(2 x^{2} - 1 \right)}}$$

2. Let $u = \cos{\left(2 x^{2} - 1 \right)}$.

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

4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(2 x^{2} - 1 \right)}$:

   1. Let $u = 2 x^{2} - 1$.

   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 $\frac{d}{d x} \left(2 x^{2} - 1\right)$:

      1. Differentiate $2 x^{2} - 1$ 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: $x^{2}$ goes to $2 x$

            So, the result is: $4 x$

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

         The result is: $4 x$

      The result of the chain rule is:

      $$- 4 x \sin{\left(2 x^{2} - 1 \right)}$$

   The result of the chain rule is:

   $$\frac{4 x \sin{\left(2 x^{2} - 1 \right)}}{\cos^{2}{\left(2 x^{2} - 1 \right)}}$$

5. Now simplify:

   $$\frac{4 x \sin{\left(2 x^{2} - 1 \right)}}{\cos^{2}{\left(2 x^{2} - 1 \right)}}$$

The answer is:

$$\frac{4 x \sin{\left(2 x^{2} - 1 \right)}}{\cos^{2}{\left(2 x^{2} - 1 \right)}}$$