If `y = x^(4) + 2 x^(2) + 3x + 1`, then `(dy)/(dx)` at x = 1 is

Options

A. 10
B. 11
C. 12
D. 7

Answer

To find \(\frac{dy}{dx}\) at \(x = 1\) for the function \(y = x^4 + 2x^2 + 3x + 1\), we will follow these steps:

### Step 1: Differentiate the function
We start with the function:
\[ y = x^4 + 2x^2 + 3x + 1 \]
Now, we differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = \frac{d}{dx}(x^4) + \frac{d}{dx}(2x^2) + \frac{d}{dx}(3x) + \frac{d}{dx}(1) \]

### Step 2: Apply the power rule
Using the power rule of differentiation, we find:
- The derivative of \(x^4\) is \(4x^3\).
- The derivative of \(2x^2\) is \(2 \cdot 2x^{2-1} = 4x\).
- The derivative of \(3x\) is \(3\).
- The derivative of a constant (1) is \(0\).

So, we can write:
\[ \frac{dy}{dx} = 4x^3 + 4x + 3 \]

### Step 3: Substitute \(x = 1\)
Now, we need to evaluate \(\frac{dy}{dx}\) at \(x = 1\):
\[ \frac{dy}{dx}\bigg|_{x=1} = 4(1)^3 + 4(1) + 3 \]

### Step 4: Calculate the values
Calculating each term:
- \(4(1)^3 = 4\)
- \(4(1) = 4\)
- \(3 = 3\)

Adding these together:
\[ \frac{dy}{dx}\bigg|_{x=1} = 4 + 4 + 3 = 11 \]

### Final Answer
Thus, \(\frac{dy}{dx}\) at \(x = 1\) is:
\[ \boxed{11} \]

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