Derivative of y=(((x^5)/8)-((x^3)/4)+(x^2))-lnx/2

1. Differentiate $\left(x^{2} + \left(\frac{x^{5}}{8} - \frac{x^{3}}{4}\right)\right) - \frac{\log{\left(x \right)}}{2}$ term by term:

   1. Differentiate $x^{2} + \left(\frac{x^{5}}{8} - \frac{x^{3}}{4}\right)$ term by term:

      1. Differentiate $\frac{x^{5}}{8} - \frac{x^{3}}{4}$ 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: $x^{5}$ goes to $5 x^{4}$

            So, the result is: $\frac{5 x^{4}}{8}$

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

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

            So, the result is: $- \frac{3 x^{2}}{4}$

         The result is: $\frac{5 x^{4}}{8} - \frac{3 x^{2}}{4}$

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

      The result is: $\frac{5 x^{4}}{8} - \frac{3 x^{2}}{4} + 2 x$

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

      1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$.

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

   The result is: $\frac{5 x^{4}}{8} - \frac{3 x^{2}}{4} + 2 x - \frac{1}{2 x}$

2. Now simplify:

   $$\frac{x^{2} \left(5 x^{3} - 6 x + 16\right) - 4}{8 x}$$

The answer is:

$$\frac{x^{2} \left(5 x^{3} - 6 x + 16\right) - 4}{8 x}$$