Differentiate w.r.t. as indicated :
`sinx^(2)" w.r.t. "x^(3)`

Answer

To differentiate \( \sin(x^2) \) with respect to \( x^3 \), we can use the chain rule and the concept of implicit differentiation. Here’s a step-by-step solution:

### Step 1: Set up the differentiation
We want to differentiate \( \sin(x^2) \) with respect to \( x^3 \). We can express this as:
\[ \frac{d}{dx^3}(\sin(x^2)) \]

### Step 2: Use the chain rule
Using the chain rule, we can rewrite this as:
\[ \frac{d}{dx^3}(\sin(x^2)) = \frac{d}{dx}(\sin(x^2)) \cdot \frac{dx}{dx^3} \]

### Step 3: Differentiate \( \sin(x^2) \)
Now, we need to differentiate \( \sin(x^2) \) with respect to \( x \):
\[ \frac{d}{dx}(\sin(x^2)) = \cos(x^2) \cdot \frac{d}{dx}(x^2) = \cos(x^2) \cdot 2x \]

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

### Step 5: Combine the results
Now we can substitute back into our expression:
\[ \frac{d}{dx^3}(\sin(x^2)) = \frac{\cos(x^2) \cdot 2x}{3x^2} \]

### Step 6: Simplify the expression
We can simplify the expression:
\[ \frac{d}{dx^3}(\sin(x^2)) = \frac{2 \cos(x^2)}{3x} \]

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