Question

How do I find the derivative of ln(8-e^(-x))?

Answer 1

To find the derivative of the function \( y = \ln(8 - e^{-x}) \), you'll apply the chain rule. The chain rule states that if you have a function of a function, its derivative is the derivative of the outer function times the derivative of the inner function.

1. **Identify the outer and inner functions:**
- Outer function: \( u = \ln(v) \) where \( v = 8 - e^{-x} \)
- Inner function: \( v = 8 - e^{-x} \)

2. **Differentiate the outer function with respect to \( v \):**
\[
\frac{du}{dv} = \frac{1}{v}
\]

3. **Differentiate the inner function with respect to \( x \):**
\[
\frac{dv}{dx} = 0 - \frac{d}{dx}(e^{-x}) = -(-e^{-x}) = e^{-x}
\]

4. **Apply the chain rule:**
\[
\frac{dy}{dx} = \frac{du}{dv} \cdot \frac{dv}{dx} = \frac{1}{v} \cdot e^{-x}
\]

5. **Substitute \( v \) back in:**
\[
\frac{dy}{dx} = \frac{e^{-x}}{8 - e^{-x}}
\]

Thus, the derivative of \( \ln(8 - e^{-x}) \) is:
\[
\frac{dy}{dx} = \frac{e^{-x}}{8 - e^{-x}}
\]