Differentiate the function.
yequals=left parenthesis 5 x minus 18 right parenthesis Superscript 5(5x−18)5

The function you want to differentiate is:

\[
y = (5x - 18)^5
\]

To differentiate this, we apply the **chain rule**, which states:

\[
\frac{d}{dx}\left[f(g(x))\right] = f'(g(x)) \cdot g'(x)
\]

Here, the outer function is \( f(u) = u^5 \) and the inner function is \( g(x) = 5x - 18 \).

### Steps:

1. Differentiate the outer function \( u^5 \) with respect to \( u \):
   \[
   \frac{d}{du}\left( u^5 \right) = 5u^4
   \]

2. Replace \( u \) with \( 5x - 18 \):
   \[
   5(5x - 18)^4
   \]

3. Now differentiate the inner function \( g(x) = 5x - 18 \):
   \[
   \frac{d}{dx}(5x - 18) = 5
   \]

4. Finally, multiply the two results:
   \[
   y' = 5(5x - 18)^4 \cdot 5
   \]

Thus, the derivative is:

\[
y' = 25(5x - 18)^4
\]

Would you like further details or have any questions?

Here are five related questions to expand on this topic:

1. How would you differentiate \( y = (3x^2 - 7)^6 \)?
2. Can you explain the chain rule in more detail?
3. What happens if the inner function includes a trigonometric term?
4. How do you apply the product rule along with the chain rule?
5. How would implicit differentiation work on a similar problem?

**Tip:** When applying the chain rule, always identify the inner and outer functions first, then differentiate each step by step.

Differentiate the function y = (5x - 18)^5

This page explains how to differentiate the function y = (5x - 18)^5 using the chain rule. The solution includes a step-by-step breakdown of applying the chain rule, focusing on both the outer and inner functions. Suitable for high school calculus students or those learning introductory calculus.

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