$f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{\left(2(x+h)^{6}+\frac{3}{(x+h)^{5}}+\frac{\sqrt{(x+h)}}{h}-9\right)-\left(2 x^{5}+\frac{3}{x^{5}}+\frac{\sqrt{x}}{4}-9\right)}{h}$

Step 1 ：First, we need to simplify the expression inside the limit. We can do this by expanding the terms and combining like terms.

Step 2 ：\(=\lim _{h \rightarrow 0} \frac{2(x^6+6x^5h+15x^4h^2+20x^3h^3+15x^2h^4+6xh^5+h^6)+\frac{3}{(x^5-5x^4h+10x^3h^2-10x^2h^3+5xh^4-h^5)}+\frac{\sqrt{x+h}}{h}-9-2x^5-\frac{3}{x^5}-\frac{\sqrt{x}}{4}+9}{h}\)

Step 3 ：Next, we can cancel out the terms that are the same in the numerator.

Step 4 ：\(=\lim _{h \rightarrow 0} \frac{2(6x^5h+15x^4h^2+20x^3h^3+15x^2h^4+6xh^5+h^6)+\frac{3}{x^5}(1-\frac{5x^4h}{x^5}+\frac{10x^3h^2}{x^5}-\frac{10x^2h^3}{x^5}+\frac{5xh^4}{x^5}-\frac{h^5}{x^5})+\frac{\sqrt{x+h}}{h}-\frac{\sqrt{x}}{4}}{h}\)

Step 5 ：Then, we can simplify the expression by dividing each term in the numerator by h.

Step 6 ：\(=\lim _{h \rightarrow 0} 2(6x^5+15x^4h+20x^3h^2+15x^2h^3+6xh^4+h^5)+\frac{3}{x^5}(1-\frac{5x^4}{x}+\frac{10x^3h}{x^2}-\frac{10x^2h^2}{x^3}+\frac{5xh^3}{x^4}-\frac{h^4}{x^5})+\sqrt{x+h}-\frac{\sqrt{x}}{4h}\)

Step 7 ：Now, we can take the limit as h approaches 0.

Step 8 ：\(=2(6x^5)+\frac{3}{x^5}(1-\frac{5x^4}{x})+\sqrt{x}-\frac{\sqrt{x}}{4}\times0\)

Step 9 ：Finally, we simplify the expression to get the derivative of f.

Step 10 ：\(=\boxed{12x^5-\frac{15}{x}+\sqrt{x}}\)