Differentiate the function.
yequals=left parenthesis 6 minus x right parenthesis Superscript 66(6−x)66
Question content area bottom
Part 1
StartFraction dy Over dx EndFraction
dy
dxequals=enter your response here

To differentiate the function \( y = (6 - x)^{66} \), we will use the **chain rule**. The chain rule is useful when differentiating composite functions.

### Step-by-step differentiation:

Given:
\[
y = (6 - x)^{66}
\]

1. **Differentiate the outer function** (which is the power function). Using the power rule \( \frac{d}{dx} [u^n] = n u^{n-1} \), where \( u = (6 - x) \) and \( n = 66 \):
   \[
   \frac{dy}{dx} = 66 \cdot (6 - x)^{65}
   \]

2. **Differentiate the inner function** (which is \( u = 6 - x \)). The derivative of \( 6 - x \) with respect to \( x \) is:
   \[
   \frac{d}{dx}(6 - x) = -1
   \]

3. **Multiply the results** from both steps using the chain rule:
   \[
   \frac{dy}{dx} = 66 \cdot (6 - x)^{65} \cdot (-1)
   \]

Thus, the derivative is:
\[
\frac{dy}{dx} = -66 \cdot (6 - x)^{65}
\]

Would you like further details on the chain rule or any specific step?

Here are 5 related questions to further your understanding:

1. How does the power rule apply to basic polynomial functions?
2. Can you differentiate \( y = (2 - x)^{100} \) using the chain rule?
3. What is the significance of the chain rule in calculus?
4. How do you differentiate a product of two functions?
5. How do you find higher-order derivatives for \( (6 - x)^{66} \)?

**Tip**: Always ensure that you differentiate both the outer and inner functions when applying the chain rule.

Differentiate the function y = (6 - x)^{66}.

Learn how to differentiate the function y = (6 - x)^66 using the chain rule. This step-by-step solution covers key calculus concepts such as the power rule and chain rule, providing a clear explanation for advanced students in Grades 11-12.

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