Obtain the derivative of `sqrt(1+x^(3)).`

Options

A. `(1)/(2)x^(2)(1+x^(3))^(-(1)/(2))`
B. `(3)/(2)x^(2)(1+x^(3))^(-(1)/(2))`
C. `(1)/(2)x^(2)(1+x^(1))^(-(1)/(2))`
D. `(3)/(2)x^(2)(1+x^(3))^(-(3)/(2))`

Explanation

To find the derivative of \( y = \sqrt{1 + x^3} \), we will use the chain rule. Here’s a step-by-step solution:

### Step 1: Rewrite the function
We start with the function:
\[ y = \sqrt{1 + x^3} \]
We can rewrite the square root in exponent form:
\[ y = (1 + x^3)^{1/2} \]

### Step 2: Apply the chain rule
To differentiate \( y \) with respect to \( x \), we apply the chain rule. The chain rule states that if \( y = f(g(x)) \), then:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]
In our case, let \( u = 1 + x^3 \). Then, we have:
\[ y = u^{1/2} \]

### Step 3: Differentiate the outer function
Now, we differentiate the outer function \( y = u^{1/2} \):
\[ \frac{dy}{du} = \frac{1}{2} u^{-1/2} \]

### Step 4: Differentiate the inner function
Next, we differentiate the inner function \( u = 1 + x^3 \):
\[ \frac{du}{dx} = 0 + 3x^2 = 3x^2 \]

### Step 5: Combine the derivatives
Now, we can combine these results using the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \frac{1}{2} u^{-1/2} \cdot 3x^2 \]

### Step 6: Substitute back for \( u \)
Now we substitute back \( u = 1 + x^3 \):
\[ \frac{dy}{dx} = \frac{1}{2} (1 + x^3)^{-1/2} \cdot 3x^2 \]

### Step 7: Simplify the expression
This simplifies to:
\[ \frac{dy}{dx} = \frac{3x^2}{2\sqrt{1 + x^3}} \]

### Final Answer
Thus, the derivative of \( y = \sqrt{1 + x^3} \) is:
\[ \frac{dy}{dx} = \frac{3x^2}{2\sqrt{1 + x^3}} \]

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