To find $f'(x)$ for the given function $f(x) = (\ln x)^8$, we need to differentiate it with respect to $x$.

Step 1: Differentiate $f(x) = (\ln x)^8$

We can apply the chain rule here. Let $u = \ln x$, so $f(x) = u^8$.

1. Differentiate $u^8$ with respect to $u$: $\frac{d}{du}(u^8) = 8u^7$

2. Differentiate $u = \ln x$ with respect to $x$: $\frac{du}{dx} = \frac{1}{x}$

Now, using the chain rule: $f'(x) = 8(\ln x)^7 \cdot \frac{1}{x} = \frac{8(\ln x)^7}{x}$

So, we have: $f'(x) = \frac{8(\ln x)^7}{x}$

Step 2: Evaluate $f'(e^4)$

To find $f'(e^4)$, substitute $x = e^4$ into the derivative:

1. Calculate $\ln(e^4)$: $\ln(e^4) = 4$

2. Substitute into $f'(x)$: $f'(e^4) = \frac{8(4)^7}{e^4}$

3. Simplify $4^7$: $4^7 = 16384$

4. Substitute the values: $f'(e^4) = \frac{8 \times 16384}{e^4} = \frac{131072}{e^4}$

Final Answer

• $f'(x) = \frac{8(\ln x)^7}{x}$
• $f'(e^4) = \frac{131072}{e^4}$

Would you like a more detailed explanation of any part?

Here are some related questions for further exploration:

1. How would the derivative change if the exponent were different, say $(\ln x)^n$?
2. How can we use logarithmic differentiation to simplify functions with exponents?
3. What is the process for differentiating composite functions like $(\ln x)^8$?
4. How would you interpret the derivative $f'(x)$ in terms of the rate of change of $f(x)$?
5. Can we use approximation methods to evaluate $f'(e^4)$ without an exact answer?

Tip: The chain rule is essential for differentiating nested functions; practice identifying inner and outer functions to apply it effectively.