Differentiate the function.
f(s) =
4
(4s7 + 2)8

To differentiate the function

\[
f(s) = 4 \left(4s^7 + 2\right)^8
\]

we can apply the **chain rule**.

### Steps:
1. **Factor out the constant**: The factor of 4 is a constant, so it can be left outside the derivative.
2. **Apply the chain rule**: The outer function is \( \left(4s^7 + 2\right)^8 \), and the inner function is \( 4s^7 + 2 \).

#### Step 1: Differentiate the outer function \( (u^8) \) with respect to \( u = 4s^7 + 2 \):
\[
\frac{d}{du} \left( u^8 \right) = 8u^7
\]
So, applying this to \( u = 4s^7 + 2 \):
\[
\frac{d}{ds} \left( (4s^7 + 2)^8 \right) = 8(4s^7 + 2)^7 \cdot \frac{d}{ds}(4s^7 + 2)
\]

#### Step 2: Differentiate the inner function \( 4s^7 + 2 \) with respect to \( s \):
\[
\frac{d}{ds}(4s^7 + 2) = 28s^6
\]

#### Step 3: Combine everything:
\[
\frac{d}{ds} f(s) = 4 \times 8 \times (4s^7 + 2)^7 \times 28s^6
\]

#### Step 4: Simplify:
\[
\frac{d}{ds} f(s) = 4 \times 8 \times 28s^6 \times (4s^7 + 2)^7
\]
\[
\frac{d}{ds} f(s) = 896s^6 (4s^7 + 2)^7
\]

Thus, the derivative is:

\[
\frac{d}{ds} f(s) = 896s^6 (4s^7 + 2)^7
\]

Would you like a further breakdown or clarification of any steps?

### Related Questions:
1. How do you apply the chain rule in more complex functions?
2. What happens if the power is negative or fractional in chain rule problems?
3. Can you differentiate composite functions involving trigonometric terms?
4. How would you integrate a function like \( f(s) = 4 (4s^7 + 2)^8 \)?
5. What are the main rules for differentiation, and when do you apply them?

**Tip**: Always identify inner and outer functions clearly when using the chain rule to avoid mistakes!

Differentiate the function f(s) = 4(4s^7 + 2)^8.

Learn how to differentiate the function f(s) = 4(4s^7 + 2)^8 by applying the chain rule and the power rule. This step-by-step solution is perfect for calculus students in Grades 11-12, covering essential differentiation techniques.

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