To differentiate the function $(x - 3)^5$ using the chain rule, let's proceed step-by-step:

Step 1: Identify the Outer and Inner Functions

The given function is $(x - 3)^5$.

• Outer function: $u^5$, where $u = (x - 3)$
• Inner function: $u = (x - 3)$

Step 2: Differentiate the Outer Function

Using the chain rule formula, which is:

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

Let’s differentiate the outer function $u^5$ with respect to $u$:

$\frac{d}{du}(u^5) = 5u^4$

Step 3: Differentiate the Inner Function

Now, differentiate the inner function $u = (x - 3)$ with respect to $x$:

$\frac{d}{dx}(x - 3) = 1$

Step 4: Apply the Chain Rule

Combine the derivatives of the outer and inner functions:

$\frac{d}{dx}\left[(x - 3)^5\right] = 5(x - 3)^4 \cdot 1$

Step 5: Simplify the Expression

The derivative simplifies to:

$\frac{d}{dx}\left[(x - 3)^5\right] = 5(x - 3)^4$

Final Answer:

$\frac{d}{dx}\left[(x - 3)^5\right] = 5(x - 3)^4$

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How do you differentiate a function with multiple nested functions using the chain rule?
2. What is the chain rule and when is it applicable?
3. How would you differentiate $(2x + 1)^3$ using the chain rule?
4. Can you explain how the power rule and chain rule work together in this example?
5. What happens if we differentiate $(x - 3)^5$ using the product rule instead of the chain rule?

Tip: When using the chain rule, always identify the outer and inner functions first, as it simplifies the differentiation process.