Derivative of 5*x^2-2/sqrt(x)+sin(pi/4)

1. Differentiate $\left(5 x^{2} - \frac{2}{\sqrt{x}}\right) + \sin{\left(\frac{\pi}{4} \right)}$ term by term:

   1. Differentiate $5 x^{2} - \frac{2}{\sqrt{x}}$ 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: $10 x$

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

         1. Let $u = \sqrt{x}$.

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

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{x}$:

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

            The result of the chain rule is:

            $$- \frac{1}{2 x^{\frac{3}{2}}}$$

         So, the result is: $\frac{1}{x^{\frac{3}{2}}}$

      The result is: $10 x + \frac{1}{x^{\frac{3}{2}}}$

   2. Let $u = \frac{\pi}{4}$.

   3. The derivative of sine is cosine:

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

   4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\pi}{4}$:

      1. The derivative of the constant $\frac{\pi}{4}$ is zero.

      The result of the chain rule is:

      $$0$$

   The result is: $10 x + \frac{1}{x^{\frac{3}{2}}}$

The answer is:

$$10 x + \frac{1}{x^{\frac{3}{2}}}$$