To find the derivative $f'(x)$ of the function $f(x) = \frac{1}{5 - 6e^x}$, we can use the chain rule and the quotient rule. Let's go through it step-by-step.

1. Rewrite $f(x)$: $f(x) = (5 - 6e^x)^{-1}$

2. Differentiate using the chain rule: For $f(x) = (5 - 6e^x)^{-1}$, let $u = 5 - 6e^x$, so $f(x) = u^{-1}$. Then $f'(x) = -u^{-2} \cdot u'$.

3. Differentiate $u = 5 - 6e^x$: $u' = -6e^x$

4. Combine the results: $f'(x) = -\frac{-6e^x}{(5 - 6e^x)^2}$

5. Simplify: $f'(x) = \frac{6e^x}{(5 - 6e^x)^2}$

So, the derivative is: $f'(x) = \frac{6e^x}{(5 - 6e^x)^2}$

Would you like additional steps or have any questions?

Here are 5 related questions you might find helpful:

1. How does the chain rule apply when differentiating functions with exponentials?
2. What is the general form of the quotient rule in calculus?
3. How can implicit differentiation be applied to functions like this?
4. What other types of functions have similar derivatives involving the chain rule?
5. How does the presence of an exponential term affect the shape of a function's graph?

Tip: When differentiating a fraction, rewrite the denominator in a simpler form when possible to make chain rule applications easier.