Find *f '*(*x*) for the function

*f*(*x*) =

1

5 − 6*ex*

.

*f '*(*x*) =

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.

Find f'(x) for the function f(x) = 1 / (5 - 6e^x).

This page provides a step-by-step solution to find the derivative f'(x) of the function f(x) = 1 / (5 - 6e^x) using the chain rule and quotient rule. Explore essential calculus concepts, including differentiation of exponential functions and application of the chain rule.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation