Differentiate: `sinx^(2)` with respect to `x^(3)`.

Answer

To differentiate \( \sin(x^2) \) with respect to \( x^3 \), we will follow these steps:

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

We need to find \( \frac{du}{dv} \).

### Step 2: Use the Chain Rule
Using the chain rule, we can express \( \frac{du}{dv} \) as:
\[ \frac{du}{dv} = \frac{du/dx}{dv/dx} \]

### Step 3: Differentiate \( u \) with respect to \( x \)
Now, we need to find \( \frac{du}{dx} \):
\[ \frac{du}{dx} = \frac{d}{dx}(\sin(x^2)) \]
Using the chain rule:
\[ \frac{du}{dx} = \cos(x^2) \cdot \frac{d}{dx}(x^2) = \cos(x^2) \cdot 2x \]

### Step 4: Differentiate \( v \) with respect to \( x \)
Next, we find \( \frac{dv}{dx} \):
\[ \frac{dv}{dx} = \frac{d}{dx}(x^3) = 3x^2 \]

### Step 5: Substitute into the Chain Rule Expression
Now we substitute \( \frac{du}{dx} \) and \( \frac{dv}{dx} \) into the expression for \( \frac{du}{dv} \):
\[ \frac{du}{dv} = \frac{\cos(x^2) \cdot 2x}{3x^2} \]

### Step 6: Simplify the Expression
We can simplify this expression:
\[ \frac{du}{dv} = \frac{2 \cos(x^2)}{3x} \]

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

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