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The given problem asks us to rewrite the function \( y = e^{-17x} \) in the form \( y = f(u) \) and \( u = g(x) \), then find \( \frac{dy}{dx} \) as a function of \( x \).

### Step 1: Express the function in terms of \( y = f(u) \) and \( u = g(x) \)

We start with:
\[
y = e^{-17x}
\]

Now, let's set:
\[
u = g(x) = -17x
\]

This allows us to write:
\[
y = f(u) = e^u
\]

Thus, we have:
\[
y = f(u), \quad u = g(x)
\]

### Step 2: Use the chain rule to differentiate

We need to find \( \frac{dy}{dx} \). Using the chain rule:
\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}
\]

From the function \( y = f(u) = e^u \), we know:
\[
\frac{dy}{du} = e^u
\]

From \( u = -17x \), we have:
\[
\frac{du}{dx} = -17
\]

Now, substituting these into the chain rule formula:
\[
\frac{dy}{dx} = e^u \cdot (-17)
\]

Since \( u = -17x \), substitute back:
\[
\frac{dy}{dx} = e^{-17x} \cdot (-17)
\]

Thus, the derivative is:
\[
\frac{dy}{dx} = -17e^{-17x}
\]

### Final Answer:
\[
\frac{dy}{dx} = -17e^{-17x}
\]

Would you like further details or explanations on any step? Here are 5 related questions to explore the topic further:

1. How is the chain rule applied to composite functions in differentiation?
2. What is the general formula for differentiating exponential functions?
3. How can we rewrite other exponential functions like \( y = e^{kx} \) for different values of \( k \)?
4. What happens if the exponent in an exponential function involves multiple variables?
5. How does the product or quotient rule interact with the chain rule?

**Tip:** When dealing with composite functions, always identify the inner and outer functions clearly to apply the chain rule efficiently.

Write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x.
Given: y = e^(-17x)

Learn how to apply the chain rule to differentiate the exponential function y = e^(-17x) with a step-by-step solution. Explore key calculus concepts such as the chain rule and exponential functions. Suitable for students in Grades 10-12.

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