What is the derivation of `x^(3)` w.r.t. `x^(2)` ?

Options

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

Answer

To find the derivative of \( x^3 \) with respect to \( x^2 \), we can follow these steps:

### Step 1: Define the Functions
Let:
- \( y = x^3 \)
- \( z = x^2 \)

We need to find \( \frac{dy}{dz} \).

### Step 2: Differentiate \( y \) with respect to \( x \)
To find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{d}{dx}(x^3) = 3x^{3-1} = 3x^2 \]

### Step 3: Differentiate \( z \) with respect to \( x \)
To find \( \frac{dz}{dx} \):
\[ \frac{dz}{dx} = \frac{d}{dx}(x^2) = 2x^{2-1} = 2x \]

### Step 4: Use the Chain Rule to Find \( \frac{dy}{dz} \)
Using the chain rule, we have:
\[ \frac{dy}{dz} = \frac{dy}{dx} \div \frac{dz}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dz} = \frac{3x^2}{2x} \]

### Step 5: Simplify the Expression
Now, simplify \( \frac{3x^2}{2x} \):
\[ \frac{dy}{dz} = \frac{3}{2} \cdot \frac{x^2}{x} = \frac{3}{2} x \]

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

---