Derivative of y=x^3-2x^2+(sqrt(x)+7)

1. Differentiate $\left(\sqrt{x} + 7\right) + \left(x^{3} - 2 x^{2}\right)$ term by term:

   1. Differentiate $x^{3} - 2 x^{2}$ term by term:

      1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$

      2. 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: $- 4 x$

      The result is: $3 x^{2} - 4 x$

   2. Differentiate $\sqrt{x} + 7$ term by term:

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

      2. The derivative of the constant $7$ is zero.

      The result is: $\frac{1}{2 \sqrt{x}}$

   The result is: $3 x^{2} - 4 x + \frac{1}{2 \sqrt{x}}$

The answer is:

$$3 x^{2} - 4 x + \frac{1}{2 \sqrt{x}}$$