To find the derivative of the function \( f(x) = t^{-3} \), we can use the power rule. The power rule states that if \( f(x) = t^n \), then the derivative is:

\[
f'(x) = n \cdot t^{n-1}
\]

### Step 1: Identify the exponent
Here, \( f(x) = t^{-3} \). So, \( n = -3 \).

### Step 2: Apply the power rule
Using the power rule:

\[
f'(x) = -3 \cdot t^{-3-1} = -3 \cdot t^{-4}
\]

### Step 3: Simplify the expression
The derivative of \( f(x) = t^{-3} \) is:

\[
f'(x) = -\frac{3}{t^4}
\]

So, the derivative is \( f'(x) = -\frac{3}{t^4} \).

Would you like more details or have any questions?

### Related Questions:
1. How do you find the derivative of a function with a negative exponent?
2. What is the derivative of \( t^{-2} \)?
3. How do you differentiate \( t^{\frac{1}{2}} \) using the power rule?
4. How does the power rule apply to fractional exponents?
5. What are the steps to differentiate a function like \( t^n \) where \( n \) is a constant?

### Tip:
When using the power rule, be careful with negative exponents—they often result in derivatives that are expressed as fractions.

Learn how to find the derivative of the function f(x) = t^-3 using the power rule. Detailed step-by-step solution with examples and related questions on derivatives with negative exponents. Suitable for Grades 11-12.

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