Derivative of y=tan²(x²+5x-2)

1. Let $u = \tan{\left(\left(x^{2} + 5 x\right) - 2 \right)}$.

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(\left(x^{2} + 5 x\right) - 2 \right)}$:

   1. Rewrite the function to be differentiated:

      $$\tan{\left(\left(x^{2} + 5 x\right) - 2 \right)} = \frac{\sin{\left(x^{2} + 5 x - 2 \right)}}{\cos{\left(x^{2} + 5 x - 2 \right)}}$$

   2. Apply the quotient rule, which is:

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

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

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

      1. Let $u = x^{2} + 5 x - 2$.

      2. The derivative of sine is cosine:

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 5 x - 2\right)$:

         1. Differentiate $x^{2} + 5 x - 2$ term by term:

            1. The derivative of the constant $-2$ is zero.

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

            3. 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 is: $2 x + 5$

         The result of the chain rule is:

         $$\left(2 x + 5\right) \cos{\left(x^{2} + 5 x - 2 \right)}$$

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

      1. Let $u = x^{2} + 5 x - 2$.

      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 $\frac{d}{d x} \left(x^{2} + 5 x - 2\right)$:

         1. Differentiate $x^{2} + 5 x - 2$ term by term:

            1. The derivative of the constant $-2$ is zero.

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

            3. 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 is: $2 x + 5$

         The result of the chain rule is:

         $$- \left(2 x + 5\right) \sin{\left(x^{2} + 5 x - 2 \right)}$$

      Now plug in to the quotient rule:

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

   The result of the chain rule is:

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

4. Now simplify:

   $$\frac{\left(4 x + 10\right) \tan{\left(x^{2} + 5 x - 2 \right)}}{\cos^{2}{\left(x^{2} + 5 x - 2 \right)}}$$

The answer is:

$$\frac{\left(4 x + 10\right) \tan{\left(x^{2} + 5 x - 2 \right)}}{\cos^{2}{\left(x^{2} + 5 x - 2 \right)}}$$