Find the derivative of $f(x)=2 x^{4} 5^{9 x+1}$
$f^{\prime}(x)=72 x^{3} \cdot 5^{9 x+1} \ln 5$
$f^{\prime}(x)=8 x^{3} \cdot 5^{9 x+1}+18 x^{4} \cdot 5^{9 x+1} \ln 5$
$f^{\prime}(x)=8 x^{3} \cdot 5^{9 x+1}+18 x^{4} \cdot 5^{9 x+1}$
$f^{\prime}(x)=8 x^{3} \cdot 5^{9 x+1} \ln 5$

Step 1 ：Given the function \(f(x)=2 x^{4} 5^{9 x+1}\)

Step 2 ：This function is a product of two functions, \(2x^4\) and \(5^{9x+1}\). The derivative of this function can be found using the product rule, which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.

Step 3 ：The derivative of \(2x^4\) is \(8x^3\) using the power rule.

Step 4 ：The derivative of \(5^{9x+1}\) is more complex and requires the chain rule. The outer function is \(5^x\) and the inner function is \(9x+1\). The derivative of \(5^x\) is \(5^x \ln 5\) and the derivative of \(9x+1\) is \(9\). Therefore, the derivative of \(5^{9x+1}\) is \(5^{9x+1} \ln 5 \cdot 9\).

Step 5 ：Therefore, the derivative of \(f(x)=2 x^{4} 5^{9 x+1}\) is \(8x^3 \cdot 5^{9x+1} + 2x^4 \cdot 5^{9x+1} \ln 5 \cdot 9\).

Step 6 ：However, there is a mistake in the above calculation. The correct derivative of \(f(x)=2 x^{4} 5^{9 x+1}\) is \(18*5^{9x+1}*x^4*\ln(5) + 8*5^{9x+1}*x^3\).

Step 7 ：Final Answer: The derivative of the function \(f(x)=2 x^{4} 5^{9 x+1}\) is \(\boxed{18*5^{9x+1}*x^4*\ln(5) + 8*5^{9x+1}*x^3}\).