Derivative of x*sqrt(x)^2-x+1

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

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

      1. Apply the product rule:

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

         $f{\left(x \right)} = x$; to find $\frac{d}{d x} f{\left(x \right)}$:

         1. Apply the power rule: $x$ goes to $1$

         $g{\left(x \right)} = \left(\sqrt{x}\right)^{2}$; to find $\frac{d}{d x} g{\left(x \right)}$:

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

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

         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:

            $$1$$

         The result is: $\left(\sqrt{x}\right)^{2} + x$

      2. 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: $-1$

      The result is: $\left(\sqrt{x}\right)^{2} + x - 1$

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

   The result is: $\left(\sqrt{x}\right)^{2} + x - 1$

2. Now simplify:

   $$2 x - 1$$

The answer is:

$$2 x - 1$$