Derivative of y(x)=1/4x^3+1/x^2-3/5x+1/6

1. Differentiate $\left(- \frac{3 x}{5} + \left(\frac{x^{3}}{4} + \frac{1}{x^{2}}\right)\right) + \frac{1}{6}$ term by term:

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

      1. Differentiate $\frac{x^{3}}{4} + \frac{1}{x^{2}}$ 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: $x^{3}$ goes to $3 x^{2}$

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

         2. Let $u = x^{2}$.

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

         4. Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{2}$:

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

            The result of the chain rule is:

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

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

      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: $- \frac{3}{5}$

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

   2. The derivative of the constant $\frac{1}{6}$ is zero.

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

The answer is:

$$\frac{3 x^{2}}{4} - \frac{3}{5} - \frac{2}{x^{3}}$$