Derivative of 1/18x^3-1/81x^5+51

1. Differentiate $\left(- \frac{x^{5}}{81} + \frac{x^{3}}{18}\right) + 51$ term by term:

   1. Differentiate $- \frac{x^{5}}{81} + \frac{x^{3}}{18}$ 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: $\frac{x^{2}}{6}$

      2. 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: $- \frac{5 x^{4}}{81}$

      The result is: $- \frac{5 x^{4}}{81} + \frac{x^{2}}{6}$

   2. The derivative of the constant $51$ is zero.

   The result is: $- \frac{5 x^{4}}{81} + \frac{x^{2}}{6}$

2. Now simplify:

   $$\frac{x^{2} \left(27 - 10 x^{2}\right)}{162}$$

The answer is:

$$\frac{x^{2} \left(27 - 10 x^{2}\right)}{162}$$