Derivative of tan(7x^2-5)

1. Rewrite the function to be differentiated:

   $$\tan{\left(7 x^{2} - 5 \right)} = \frac{\sin{\left(7 x^{2} - 5 \right)}}{\cos{\left(7 x^{2} - 5 \right)}}$$

2. Apply the quotient rule, which is:

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

   $f{\left(x \right)} = \sin{\left(7 x^{2} - 5 \right)}$ and $g{\left(x \right)} = \cos{\left(7 x^{2} - 5 \right)}$.

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

   1. Let $u = 7 x^{2} - 5$.

   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 $\frac{d}{d x} \left(7 x^{2} - 5\right)$:

      1. Differentiate $7 x^{2} - 5$ 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: $14 x$

         2. The derivative of the constant $\left(-1\right) 5$ is zero.

         The result is: $14 x$

      The result of the chain rule is:

      $$14 x \cos{\left(7 x^{2} - 5 \right)}$$

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

   1. Let $u = 7 x^{2} - 5$.

   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 $\frac{d}{d x} \left(7 x^{2} - 5\right)$:

      1. Differentiate $7 x^{2} - 5$ 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: $14 x$

         2. The derivative of the constant $\left(-1\right) 5$ is zero.

         The result is: $14 x$

      The result of the chain rule is:

      $$- 14 x \sin{\left(7 x^{2} - 5 \right)}$$

   Now plug in to the quotient rule:

   $\frac{14 x \sin^{2}{\left(7 x^{2} - 5 \right)} + 14 x \cos^{2}{\left(7 x^{2} - 5 \right)}}{\cos^{2}{\left(7 x^{2} - 5 \right)}}$

3. Now simplify:

   $$\frac{14 x}{\cos^{2}{\left(7 x^{2} - 5 \right)}}$$

The answer is: