

Find the derivative of  f left parenthesis x right parenthesis equals 2 x to the power of 4 5 to the power of 9 x plus 1 end exponent.

    f apostrophe left parenthesis x right parenthesis equals 8 x cubed times 5 to the power of 9 x plus 1 end exponent plus 18 x to the power of 4 times 5 to the power of 9 x plus 1 end exponent

    f apostrophe left parenthesis x right parenthesis equals 8 x cubed times 5 to the power of 9 x plus 1 end exponent ln 5

    f apostrophe left parenthesis x right parenthesis equals 8 x cubed times 5 to the power of 9 x plus 1 end exponent plus 18 x to the power of 4 times 5 to the power of 9 x plus 1 end exponent ln 5

    f apostrophe left parenthesis x right parenthesis equals 72 x cubed times 5 to the power of 9 x plus 1 end exponent ln 5



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.

Find the derivative of f(x) = 2x^4 * 5^(9x + 1).

Learn how to find the derivative of the function f(x) = 2x^4 * 5^(9x + 1) step-by-step using the product and chain rules. This problem involves key differentiation concepts, including the product rule, chain rule, and exponential functions. Suitable for undergraduate calculus students.

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