Derivative of y=x²sec(x)

1. Apply the product rule:

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

   $f{\left(x \right)} = x^{2}$; to find $\frac{d}{d x} f{\left(x \right)}$:

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

   $g{\left(x \right)} = \sec{\left(x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

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

2. Now simplify:

   $$\frac{x \left(x \tan{\left(x \right)} + 2\right)}{\cos{\left(x \right)}}$$

The answer is:

$$\frac{x \left(x \tan{\left(x \right)} + 2\right)}{\cos{\left(x \right)}}$$