`y=e^(sin x^(3))`

Explanation

To find the derivative of the function \( y = e^{\sin(x^3)} \), we will use the chain rule of differentiation. Here’s a step-by-step solution:

### Step 1: Identify the outer and inner functions
The function can be expressed as a composition of functions:
- Outer function: \( u = e^v \) where \( v = \sin(x^3) \)
- Inner function: \( v = \sin(x^3) \)

### Step 2: Differentiate the outer function
The derivative of \( e^v \) with respect to \( v \) is:
\[ \frac{du}{dv} = e^v \]

### Step 3: Differentiate the inner function
Next, we differentiate \( v = \sin(x^3) \) with respect to \( x \). We will again use the chain rule here:
- The derivative of \( \sin(w) \) is \( \cos(w) \) where \( w = x^3 \).
- The derivative of \( w = x^3 \) with respect to \( x \) is \( 3x^2 \).

Thus, we have:
\[ \frac{dv}{dx} = \cos(x^3) \cdot \frac{d}{dx}(x^3) = \cos(x^3) \cdot 3x^2 \]

### Step 4: Apply the chain rule
Now we can apply the chain rule:
\[ \frac{dy}{dx} = \frac{du}{dv} \cdot \frac{dv}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = e^{\sin(x^3)} \cdot \left( \cos(x^3) \cdot 3x^2 \right) \]

### Step 5: Simplify the expression
Combining everything, we get:
\[ \frac{dy}{dx} = 3x^2 \cos(x^3) e^{\sin(x^3)} \]

### Final Answer
Thus, the derivative of \( y = e^{\sin(x^3)} \) is:
\[ \frac{dy}{dx} = 3x^2 \cos(x^3) e^{\sin(x^3)} \]

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