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find the derivative of the following functions



The derivative of the function \( f(x) = (3x - 5)^5 \) can be calculated using the **chain rule**.

### Chain Rule:
If \( y = [u(x)]^n \), then the derivative is given by:

\[
\frac{dy}{dx} = n \cdot [u(x)]^{n-1} \cdot \frac{du}{dx}
\]

Here:
- \( u(x) = 3x - 5 \)
- \( n = 5 \)

Now, applying the chain rule:

1. The outer function is raised to the power 5, so the derivative of \( (3x - 5)^5 \) with respect to \( (3x - 5) \) is:
\[
5 \cdot (3x - 5)^4
\]

2. The derivative of the inner function \( 3x - 5 \) with respect to \( x \) is:
\[
\frac{d}{dx}(3x - 5) = 3
\]

Thus, multiplying both results together, the derivative is:

\[
f'(x) = 5 \cdot (3x - 5)^4 \cdot 3 = 15 \cdot (3x - 5)^4
\]

So the final result is:
\[
f'(x) = 15 \cdot (3x - 5)^4
\]

Let me know if you would like further details or have any questions!

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Here are 5 related questions you might find helpful:
1. What is the derivative of \( f(x) = (4x + 2)^6 \)?
2. Can the chain rule be used for trigonometric functions?
3. How would you apply the product rule to a function like \( f(x) = x^2 (3x - 5)^3 \)?
4. What is the second derivative of \( f(x) = (3x - 5)^5 \)?
5. How do you differentiate implicit functions involving powers?

**Tip**: Always apply the chain rule when differentiating composite functions!

Find the derivative of the function f(x) = (3x - 5)^5.

Learn how to find the derivative of the function f(x) = (3x - 5)^5 using the chain rule in calculus. This solution covers differentiation techniques for composite functions, suitable for high school students in Grades 11-12.

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