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Let's go through each problem and find the derivative \( f'(x) \) as required.

### 1. \( y = x^{18} \)
Using the power rule \( \frac{d}{dx} x^n = nx^{n-1} \):
\[
f'(x) = 18x^{17}
\]

### 2. \( y = x^3 - 6x^2 + 4x + 5 \)
Differentiating each term separately:
\[
f'(x) = 3x^2 - 12x + 4
\]
(The derivative of the constant \(5\) is \(0\).)

### 3. \( y = 2\sqrt{x} + 3\sqrt[3]{x} + 4\sqrt[4]{x} \)
Rewrite the square root and other radicals as fractional exponents:
- \( \sqrt{x} = x^{1/2} \)
- \( \sqrt[3]{x} = x^{1/3} \)
- \( \sqrt[4]{x} = x^{1/4} \)

Now differentiate using the power rule:
\[
f'(x) = 2 \cdot \frac{1}{2}x^{-\frac{1}{2}} + 3 \cdot \frac{1}{3}x^{-\frac{2}{3}} + 4 \cdot \frac{1}{4}x^{-\frac{3}{4}}
\]
Simplifying:
\[
f'(x) = x^{-\frac{1}{2}} + x^{-\frac{2}{3}} + x^{-\frac{3}{4}}
\]

### 4. \( y = 3x^{\frac{2}{3}} + 4x^{\frac{1}{2}} + 5x^{\frac{4}{5}} \)
Apply the power rule to each term:
\[
f'(x) = 3 \cdot \frac{2}{3}x^{-\frac{1}{3}} + 4 \cdot \frac{1}{2}x^{-\frac{1}{2}} + 5 \cdot \frac{4}{5}x^{-\frac{1}{5}}
\]
Simplifying:
\[
f'(x) = 2x^{-\frac{1}{3}} + 2x^{-\frac{1}{2}} + 4x^{-\frac{1}{5}}
\]

Would you like further details or clarification on any of these steps?

Here are some related questions:
1. How would the derivative change if the exponent in problem 1 was negative?
2. Can you find the second derivative of the function in problem 2?
3. What is the physical interpretation of a derivative in real-world applications?
4. How would you differentiate a function that includes a product of two terms, like \(y = x^3 \cdot e^x\)?
5. How do you apply the chain rule to more complicated functions?

**Tip:** When differentiating terms involving roots, rewrite them as exponents first, which makes applying the power rule much easier!

Find f'(x) for the following functions:
1. y = x^18
2. y = x^3 - 6x^2 + 4x + 5
3. y = 2√x + 3∛x + 4∜x
4. y = 3x^(2/3) + 4x^(1/2) + 5x^(4/5)

Step-by-step solution to find the derivatives of polynomial and radical functions, including y = x^18, y = x^3 - 6x^2 + 4x + 5, and more. Learn the application of the power rule and how to handle fractional exponents. Suitable for high school calculus students.

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