Find `dy/dx` for each of the following
`y=root(3)(x^(2)(x^(2)+3))`.

Answer

To find the derivative \( \frac{dy}{dx} \) for the function \( y = \sqrt[3]{x^2(x^2 + 3)} \), we will follow these steps:

### Step 1: Rewrite the function
First, we rewrite the function in a more manageable form:
\[ y = (x^2(x^2 + 3))^{1/3} \]

### Step 2: Simplify the expression inside the cube root
Next, we simplify the expression inside the cube root:
\[ x^2(x^2 + 3) = x^4 + 3x^2 \]
Thus, we can rewrite \( y \) as:
\[ y = (x^4 + 3x^2)^{1/3} \]

### Step 3: Apply the chain rule
To differentiate \( y \), we apply the chain rule. Let \( u = x^4 + 3x^2 \), then \( y = u^{1/3} \). According to the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]

### Step 4: Differentiate \( y \) with respect to \( u \)
Now, we differentiate \( y \) with respect to \( u \):
\[ \frac{dy}{du} = \frac{1}{3} u^{-2/3} \]

### Step 5: Differentiate \( u \) with respect to \( x \)
Next, we differentiate \( u \) with respect to \( x \):
\[ u = x^4 + 3x^2 \implies \frac{du}{dx} = 4x^3 + 6x \]

### Step 6: Combine the derivatives
Now we can combine the derivatives using the chain rule:
\[ \frac{dy}{dx} = \frac{1}{3} u^{-2/3} \cdot (4x^3 + 6x) \]

### Step 7: Substitute back for \( u \)
Substituting back \( u = x^4 + 3x^2 \):
\[ \frac{dy}{dx} = \frac{1}{3} (x^4 + 3x^2)^{-2/3} \cdot (4x^3 + 6x) \]

### Final Expression
Thus, the final expression for \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{4x^3 + 6x}{3(x^4 + 3x^2)^{2/3}} \]

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