differentiate using the chain rule and showing all working step by step, (x-3)^5

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.

Explore a detailed, step-by-step solution to differentiate (x-3)^5 using the chain rule. This solution covers essential calculus concepts, including identifying outer and inner functions, applying the power rule, and simplifying expressions. Ideal for students in Grades 11-12.

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