Differentiate.

f left parenthesis x right parenthesis equals 7 e Superscript 8 x

To differentiate the function \( f(x) = 7e^{8x} \), we apply the chain rule.

The derivative of an exponential function \( e^{u(x)} \) is given by \( \frac{d}{dx} e^{u(x)} = e^{u(x)} \cdot u'(x) \), where \( u(x) \) is the function inside the exponent.

Here, we have:
\[
f(x) = 7e^{8x}
\]

Differentiate \( f(x) \):

\[
f'(x) = 7 \cdot \frac{d}{dx} \left( e^{8x} \right)
\]

Since \( \frac{d}{dx} e^{8x} = e^{8x} \cdot 8 \), we get:

\[
f'(x) = 7 \cdot e^{8x} \cdot 8
\]

Simplifying:

\[
f'(x) = 56e^{8x}
\]

So, the derivative is:

\[
f'(x) = 56e^{8x}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the derivative of \( e^{kx} \) where \( k \) is a constant?
2. How do you apply the chain rule in differentiation?
3. Can you differentiate the function \( f(x) = 9e^{5x} \)?
4. What is the second derivative of \( f(x) = 7e^{8x} \)?
5. How do you differentiate exponential functions with respect to \( x \)?

**Tip:** Always apply the chain rule when differentiating composite functions like \( e^{g(x)} \), where \( g(x) \) is any function of \( x \).

Learn how to differentiate the function f(x) = 7e^{8x} using the chain rule in calculus. This solution demonstrates step-by-step application of the chain rule and exponential differentiation. Suitable for advanced high school students and introductory calculus courses.

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