Derivative of y=e^secx-4✓x

1. Differentiate $e^{\sec{\left(x \right)}} - 4 \sqrt{x}$ term by term:

   1. Let $u = \sec{\left(x \right)}$.

   2. The derivative of $e^{u}$ is itself.

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

      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 of the chain rule is:

      $$\frac{e^{\sec{\left(x \right)}} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$$

   4. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$

      So, the result is: $- \frac{2}{\sqrt{x}}$

   The result is: $\frac{e^{\sec{\left(x \right)}} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{2}{\sqrt{x}}$

2. Now simplify:

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

The answer is:

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