https://file.aiquickdraw.com/mathbot/file/17272812346914c12do2z.jpg

The function provided is:

\[
f(x) = 14 \cdot \frac{1}{x^3} + x^2 + 14
\]

To find the derivative \( f'(x) \), we will differentiate each term separately.

### Step 1: Differentiate \( 14 \cdot \frac{1}{x^3} \)
We can rewrite this term as \( 14x^{-3} \), then apply the power rule:

\[
\frac{d}{dx} \left( 14x^{-3} \right) = 14 \cdot (-3) \cdot x^{-4} = -42x^{-4}
\]

### Step 2: Differentiate \( x^2 \)
Using the power rule for \( x^2 \):

\[
\frac{d}{dx} \left( x^2 \right) = 2x
\]

### Step 3: Differentiate \( 14 \)
Since 14 is a constant, its derivative is:

\[
\frac{d}{dx} (14) = 0
\]

### Final Result
Now combine all the terms:

\[
f'(x) = -42x^{-4} + 2x
\]

Thus, the derivative of the function is:

\[
f'(x) = -\frac{42}{x^4} + 2x
\]

Would you like further clarification or more details on any step?

### 5 Related Questions:
1. How do you apply the power rule for derivatives?
2. What is the derivative of a constant?
3. How would you differentiate more complex functions like \( \frac{1}{x^n} \)?
4. Can you explain how the chain rule is applied in differentiation?
5. What is the physical interpretation of a derivative in real-life problems?

### Tip:
When differentiating negative exponents, always rewrite them as powers of \(x\) to easily apply the power rule.

Find the derivative of the function f(x) = 14 * (1/x^3) + x^2 + 14.

Learn how to find the derivative of the function f(x) = 14(1/x^3) + x^2 + 14 using the power rule. This solution includes step-by-step differentiation, applying the power rule, and handling constants. Suitable for calculus students in Grades 11-12.

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