f\left(x\right)=\frac{3}{\sqrt{x}}-\frac{4}{3x^3} whats the derivative

To find the derivative of the function \( f(x) = \frac{3}{\sqrt{x}} - \frac{4}{3x^3} \), we'll apply the rules of differentiation.

### Step 1: Rewrite the function
First, let's rewrite the function using exponent notation:
\[
f(x) = 3x^{-\frac{1}{2}} - \frac{4}{3}x^{-3}
\]

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

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

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

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

Alternatively, you can rewrite it back in terms of radicals if preferred:
\[
f'(x) = -\frac{3}{2\sqrt{x^3}} + \frac{4}{x^4}
\]

Would you like any further details on the differentiation process, or do you have any additional questions?

### Related Questions:
1. How do you differentiate a function with a square root?
2. What are the rules for differentiating power functions?
3. Can you explain the chain rule in differentiation?
4. How does the product rule apply to functions like this?
5. What are the common mistakes when differentiating negative exponents?

### Tip:
When differentiating functions with roots, convert them to exponent form first for easier application of the power rule.

Learn how to find the derivative of the function f(x) = 3/sqrt(x) - 4/(3x^3) using differentiation rules. Detailed step-by-step solution with explanations of applying the power rule and handling negative exponents. Suitable for college-level students and those studying calculus.

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