Find the derivative of `y = e^(x^3)`

Answer

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

### Step 1: Identify the outer and inner functions
We can identify the outer function as \( e^u \) where \( u = x^3 \).

### Step 2: Differentiate the outer function
The derivative of \( e^u \) with respect to \( u \) is \( e^u \).

### Step 3: Differentiate the inner function
Now, we need to differentiate the inner function \( u = x^3 \) with respect to \( x \). The derivative of \( x^3 \) is \( 3x^2 \).

### Step 4: Apply the chain rule
Using the chain rule, we have:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = e^{u} \cdot 3x^2 \]

### Step 5: Substitute back the inner function
Now, we substitute \( u = x^3 \) back into the equation:
\[ \frac{dy}{dx} = e^{x^3} \cdot 3x^2 \]

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

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