Find the derivative of the function.

r'(x)= (4/3x^2) + 1/x^3.3

To find the derivative of the function:

\[
r(x) = \frac{4}{3x^2} + \frac{1}{x^{3.3}}
\]

we will differentiate each term individually using the power rule for derivatives.

### Step 1: Rewrite the function
Rewriting the terms to express them with negative exponents, we have:

\[
r(x) = \frac{4}{3}x^{-2} + x^{-3.3}
\]

### Step 2: Differentiate each term

1. For the first term \(\frac{4}{3}x^{-2}\), apply the power rule \(\frac{d}{dx}[x^n] = n x^{n-1}\):

\[
\frac{d}{dx} \left( \frac{4}{3}x^{-2} \right) = \frac{4}{3} \cdot (-2)x^{-3} = -\frac{8}{3}x^{-3}
\]

2. For the second term \(x^{-3.3}\), apply the same power rule:

\[
\frac{d}{dx}(x^{-3.3}) = -3.3x^{-4.3}
\]

### Step 3: Combine the results

The derivative of the function is:

\[
r'(x) = -\frac{8}{3}x^{-3} - 3.3x^{-4.3}
\]

You can leave it in this form, or rewrite it using positive exponents:

\[
r'(x) = -\frac{8}{3x^3} - \frac{3.3}{x^{4.3}}
\]

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Would you like more details on any step or have questions?

### Related Questions:
1. How would you differentiate a product of two functions?
2. What is the derivative of \(x^{1.5}\)?
3. How do you apply the chain rule in calculus?
4. What is the second derivative of this function?
5. How would the derivative change if we had a logarithmic function instead?

### Tip:
When differentiating terms with fractions or decimals in the exponents, rewriting them in terms of negative exponents helps simplify the application of the power rule.

Find the derivative of the function. r'(x)= (4/3x^2) + 1/x^3.3

Learn how to find the derivative of the function (4/3x^2) + 1/x^3.3 using the power rule in calculus. This step-by-step solution explains how to differentiate each term individually, including terms with fractional exponents. Suitable for students in Grades 11-12.

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