`y=e^(sin x^(3)) fi nd dy/dx `

Explanation

To find the derivative \( \frac{dy}{dx} \) of the function \( y = e^{\sin(x^3)} \), we will use the chain rule of differentiation. The chain rule states that if you have a composite function \( y = f(g(x)) \), then the derivative is given by \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).

### Step-by-Step Solution:

1. **Identify the outer and inner functions**:
Here, the outer function is \( f(u) = e^u \) where \( u = \sin(x^3) \). The inner function is \( g(x) = \sin(x^3) \).

2. **Differentiate the outer function**:
The derivative of \( f(u) = e^u \) with respect to \( u \) is:
\[ f'(u) = e^u \]
Therefore, when substituting back for \( u \), we have:
\[ f'(\sin(x^3)) = e^{\sin(x^3)} \]

3. **Differentiate the inner function**:
Next, we need to differentiate \( g(x) = \sin(x^3) \). Using the chain rule again:
\[ g'(x) = \cos(x^3) \cdot \frac{d}{dx}(x^3) \]
The derivative of \( x^3 \) is \( 3x^2 \), so:
\[ g'(x) = \cos(x^3) \cdot 3x^2 \]

4. **Apply the chain rule**:
Now we can combine these results using the chain rule:
\[ \frac{dy}{dx} = f'(\sin(x^3)) \cdot g'(x) = e^{\sin(x^3)} \cdot (\cos(x^3) \cdot 3x^2) \]

5. **Final expression**:
Therefore, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 3x^2 \cos(x^3) e^{\sin(x^3)} \]

### Summary of the Solution:
\[ \frac{dy}{dx} = 3x^2 \cos(x^3) e^{\sin(x^3)} \]