Find the derivative with respect to x of the following:
`(2x +x^4)/( x^2)`

Answer

To find the derivative of the function \( y = \frac{2x + x^4}{x^2} \) with respect to \( x \), we can follow these steps:

### Step 1: Simplify the Function
First, we simplify the function by separating the terms in the numerator:

\[ y = \frac{2x}{x^2} + \frac{x^4}{x^2} \]

### Step 2: Simplify Each Term
Next, we simplify each term:

\[ y = \frac{2}{x} + x^2 \]

### Step 3: Differentiate the Function
Now we differentiate \( y \) with respect to \( x \):

\[ \frac{dy}{dx} = \frac{d}{dx}\left(\frac{2}{x}\right) + \frac{d}{dx}(x^2) \]

### Step 4: Apply the Power Rule
We can rewrite \( \frac{2}{x} \) as \( 2x^{-1} \) and then apply the power rule:

\[ \frac{dy}{dx} = 2 \cdot \frac{d}{dx}(x^{-1}) + \frac{d}{dx}(x^2) \]

Using the power rule \( \frac{d}{dx}(x^n) = n x^{n-1} \):

\[ \frac{dy}{dx} = 2 \cdot (-1)x^{-2} + 2x^{1} \]

### Step 5: Simplify the Derivative
Now we simplify the expression:

\[ \frac{dy}{dx} = -\frac{2}{x^2} + 2x \]

### Step 6: Factor Out Common Terms
We can factor out the common factor of 2:

\[ \frac{dy}{dx} = 2\left(x - \frac{1}{x^2}\right) \]

### Final Answer
Thus, the derivative of the function is:

\[ \frac{dy}{dx} = 2\left(x - \frac{1}{x^2}\right) \]

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