f\left(x\right)=3\:\sqrt[3]{x^4}-\frac{2}{\sqrt[4]{x^3}}+\frac{1}{2x^4}

To find the derivative of the function \( f(x) = 3\sqrt[3]{x^4} - \frac{2}{\sqrt[4]{x^3}} + \frac{1}{2x^4} \), we'll first rewrite the function using exponent notation, then differentiate each term.

### Step 1: Rewrite the function
Express the radicals using exponents:
\[
f(x) = 3x^{\frac{4}{3}} - 2x^{-\frac{3}{4}} + \frac{1}{2}x^{-4}
\]

### Step 2: Differentiate each term
Now, differentiate each term separately:

1. For the first term \( 3x^{\frac{4}{3}} \):
   \[
   \frac{d}{dx}\left(3x^{\frac{4}{3}}\right) = 3 \cdot \frac{4}{3} x^{\frac{4}{3} - 1} = 4x^{\frac{1}{3}}
   \]

2. For the second term \( -2x^{-\frac{3}{4}} \):
   \[
   \frac{d}{dx}\left(-2x^{-\frac{3}{4}}\right) = -2 \cdot \left(-\frac{3}{4}\right) x^{-\frac{3}{4} - 1} = \frac{3}{2}x^{-\frac{7}{4}}
   \]

3. For the third term \( \frac{1}{2}x^{-4} \):
   \[
   \frac{d}{dx}\left(\frac{1}{2}x^{-4}\right) = \frac{1}{2} \cdot (-4) x^{-5} = -2x^{-5}
   \]

### Step 3: Combine the results
So, the derivative \( f'(x) \) is:
\[
f'(x) = 4x^{\frac{1}{3}} + \frac{3}{2}x^{-\frac{7}{4}} - 2x^{-5}
\]

This expression gives the derivative of the function.

Would you like any additional explanations or have further questions?

### Related Questions:
1. How do you differentiate functions with fractional exponents?
2. What are the steps for differentiating a sum of terms?
3. Can you explain how to simplify exponents before differentiation?
4. What are the rules for differentiating negative exponents?
5. How do you handle mixed exponents (positive and negative) in differentiation?

### Tip:
When differentiating functions with fractional exponents, it’s often easier to first express everything using exponents and then apply the power rule.

Learn how to differentiate the function f(x) = 3∛(x^4) - 2/(4∛(x^3)) + 1/(2x^4) with a step-by-step solution. This problem involves differentiation of exponential and fractional exponent functions, using the power rule. Suitable for advanced high school students or college calculus students.

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