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.