`d/(dx) (e^(x^3))` is equal to

Options

A. `3xe^(x^3)`
B. `3x^(2)e^(x^3)`
C. `3x(e^(x^3))^(2)`
D. `2x^(2)e^(x^3)`

Answer

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

### Step 1: Identify the function
We have:
\[ y = e^{x^3} \]

### Step 2: Apply the chain rule
The chain rule states that if you have a composite function \( y = e^{u} \) where \( u = x^3 \), then the derivative \( \frac{dy}{dx} \) is given by:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]

### Step 3: Differentiate the outer function
The derivative of \( e^u \) with respect to \( u \) is:
\[ \frac{dy}{du} = e^{u} = e^{x^3} \]

### Step 4: Differentiate the inner function
Now we need to differentiate \( u = x^3 \) with respect to \( x \):
\[ \frac{du}{dx} = 3x^{2} \]

### Step 5: Combine the derivatives
Now we can combine the results from Steps 3 and 4:
\[ \frac{dy}{dx} = e^{x^3} \cdot 3x^{2} \]

### Step 6: Write the final answer
Thus, the derivative of \( y = e^{x^3} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 3x^{2} e^{x^3} \]

### Final Answer:
\[ \frac{d}{dx} (e^{x^3}) = 3x^{2} e^{x^3} \]

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