Using the 4 step rule, find the derivative of y=3x to the power of negative 1 minus 4x to the power of negative 2 

To differentiate the function \( y = 3x^{-1} - 4x^{-2} \) using the four-step rule, we break down the process as follows:

### Step 1: Apply the power rule to each term
The power rule states that for \( y = ax^n \), the derivative is \( \frac{d}{dx}(ax^n) = n \cdot ax^{n-1} \).

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

2. For the second term \( -4x^{-2} \), apply the power rule:
   \[
   \frac{d}{dx}(-4x^{-2}) = -2 \cdot (-4)x^{-3} = 8x^{-3}
   \]

### Step 2: Combine the derivatives
Now, combine the results of the two derivatives:
\[
\frac{dy}{dx} = -\frac{3}{x^2} + 8x^{-3}
\]

### Step 3: Simplify (if necessary)
The derivative is already in its simplest form. Thus, the final expression for the derivative is:
\[
\frac{dy}{dx} = -\frac{3}{x^2} + \frac{8}{x^3}
\]

### Step 4: Verify and interpret
The derivative has been correctly applied using the power rule for each term.

Would you like a more detailed breakdown or have any specific questions?

#### 5 Related Questions:
1. How would you find the second derivative of this function?
2. What does the derivative tell us about the behavior of the original function?
3. How do higher-order derivatives affect the shape of the graph?
4. What happens to the derivative as \( x \to 0 \)?
5. Can we apply the product or chain rule to this problem?

**Tip:** Always double-check the power rule when differentiating terms with negative exponents!

Using the 4 step rule, find the derivative of y=3x to the power of negative 1 minus 4x to the power of negative 2

Learn how to find the derivative of y = 3x^(-1) - 4x^(-2) using the power rule, with a step-by-step breakdown. This solution involves key calculus concepts and is suitable for students in Grades 10-12.

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