What is the derivative of `x^(3)` with respect to `x^(2)`?

Options

A. `3x^(2)`
B. `(3x)/(2)`
C. `x`
D. `(3)/(2)`

Explanation

To find the derivative of \( x^3 \) with respect to \( x^2 \), we can use the chain rule. Here’s the step-by-step solution:

### Step 1: Define the functions
Let:
- \( u = x^3 \)
- \( v = x^2 \)

### Step 2: Differentiate \( u \) with respect to \( x \)
To find \( \frac{du}{dx} \):
\[ \frac{du}{dx} = \frac{d}{dx}(x^3) = 3x^2 \]

### Step 3: Differentiate \( v \) with respect to \( x \)
To find \( \frac{dv}{dx} \):
\[ \frac{dv}{dx} = \frac{d}{dx}(x^2) = 2x \]

### Step 4: Use the chain rule to find \( \frac{du}{dv} \)
According to the chain rule:
\[ \frac{du}{dv} = \frac{du/dx}{dv/dx} \]
Substituting the derivatives we found:
\[ \frac{du}{dv} = \frac{3x^2}{2x} \]

### Step 5: Simplify the expression
Now simplify \( \frac{3x^2}{2x} \):
\[ \frac{du}{dv} = \frac{3}{2} \cdot \frac{x^2}{x} = \frac{3}{2} x \]

### Final Result
Thus, the derivative of \( x^3 \) with respect to \( x^2 \) is:
\[ \frac{du}{dv} = \frac{3}{2} x \]

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