The given function is $f(x) = 2x^4 \cdot 5^{9x+1}$.

Let's break down the derivative step by step:

Step 1: Use the product rule.

Since the function is the product of two terms, $2x^4$ and $5^{9x+1}$, we apply the product rule, which states:

$(fg)' = f'g + fg'$

Here, $f(x) = 2x^4$ and $g(x) = 5^{9x+1}$.

Step 2: Derivative of $2x^4$.

$f'(x) = 8x^3$

Step 3: Derivative of $5^{9x+1}$.

To differentiate $5^{9x+1}$, we use the chain rule and the fact that the derivative of $a^x$ with respect to $x$ is $a^x \ln a$. So, the derivative of $5^{9x+1}$ is:

$g'(x) = 5^{9x+1} \cdot \ln(5) \cdot 9$ $g'(x) = 9 \ln(5) \cdot 5^{9x+1}$

Step 4: Applying the product rule.

Now, using the product rule:

$f'(x) = (8x^3 \cdot 5^{9x+1}) + (2x^4 \cdot 9 \ln(5) \cdot 5^{9x+1})$

Simplifying this expression:

$f'(x) = 8x^3 \cdot 5^{9x+1} + 18x^4 \cdot \ln(5) \cdot 5^{9x+1}$

Step 5: Factoring the common terms.

Both terms have $5^{9x+1}$, so we can factor it out:

$f'(x) = 5^{9x+1} \cdot \left( 8x^3 + 18x^4 \cdot \ln(5) \right)$

Conclusion:

None of the options provided exactly matches this derivative. However, the correct form of the derivative is:

$f'(x) = 5^{9x+1} \cdot \left( 8x^3 + 18x^4 \cdot \ln(5) \right)$

Would you like further details or have any questions?

Here are 5 related questions:

1. What is the chain rule in differentiation, and how is it applied?
2. How can we factor common terms when simplifying expressions in derivatives?
3. What is the derivative of a general exponential function $a^{bx+c}$?
4. How does the product rule work for functions that are products of more than two terms?
5. Can logarithmic properties simplify the differentiation process?

Tip: When differentiating products of functions, always start by identifying which rules (like product, chain, or quotient) are needed to avoid mistakes.