If ` y= x ^(3) +5x^(2) -3x +10 ,then (d^(2)y)/(dx^(2))= `

Options

A. ` 3x^(2) +10 x `
B. ` 3x^(2) +5x`
C. ` 3x + 5`
D. ` 6x+ 10 `

Answer

To find the second derivative of the function \( y = x^3 + 5x^2 - 3x + 10 \), we will follow these steps:

### Step 1: Find the first derivative \( \frac{dy}{dx} \)

Using the power rule of differentiation, which states that \( \frac{d}{dx}(x^n) = nx^{n-1} \), we differentiate each term of the function:

1. The derivative of \( x^3 \) is \( 3x^2 \).
2. The derivative of \( 5x^2 \) is \( 10x \).
3. The derivative of \( -3x \) is \( -3 \).
4. The derivative of the constant \( 10 \) is \( 0 \).

Putting it all together, we have:
\[ \frac{dy}{dx} = 3x^2 + 10x - 3 \]

### Step 2: Find the second derivative \( \frac{d^2y}{dx^2} \)

Now we differentiate the first derivative \( \frac{dy}{dx} = 3x^2 + 10x - 3 \) again:

1. The derivative of \( 3x^2 \) is \( 6x \).
2. The derivative of \( 10x \) is \( 10 \).
3. The derivative of the constant \( -3 \) is \( 0 \).

Combining these results, we find:
\[ \frac{d^2y}{dx^2} = 6x + 10 \]

### Final Answer
Thus, the second derivative \( \frac{d^2y}{dx^2} \) is:
\[ 6x + 10 \]