Which one of the following is the second derivatives of equation below?


P = 4Q3 + 7Q2 + 2Q

To find the second derivative of the function \( P = 4Q^3 + 7Q^2 + 2Q \), we follow these steps:

### 1. First derivative \( \frac{dP}{dQ} \):

\[
\frac{dP}{dQ} = \frac{d}{dQ} \left( 4Q^3 \right) + \frac{d}{dQ} \left( 7Q^2 \right) + \frac{d}{dQ} \left( 2Q \right)
\]

- The derivative of \( 4Q^3 \) is \( 12Q^2 \).
- The derivative of \( 7Q^2 \) is \( 14Q \).
- The derivative of \( 2Q \) is \( 2 \).

So the first derivative is:

\[
\frac{dP}{dQ} = 12Q^2 + 14Q + 2
\]

### 2. Second derivative \( \frac{d^2P}{dQ^2} \):

Now, we differentiate the first derivative with respect to \( Q \):

\[
\frac{d^2P}{dQ^2} = \frac{d}{dQ} \left( 12Q^2 \right) + \frac{d}{dQ} \left( 14Q \right) + \frac{d}{dQ} \left( 2 \right)
\]

- The derivative of \( 12Q^2 \) is \( 24Q \).
- The derivative of \( 14Q \) is \( 14 \).
- The derivative of \( 2 \) is \( 0 \) (as constants have a derivative of zero).

So the second derivative is:

\[
\frac{d^2P}{dQ^2} = 24Q + 14
\]

### Final Answer:
The second derivative of \( P = 4Q^3 + 7Q^2 + 2Q \) is:

\[
\frac{d^2P}{dQ^2} = 24Q + 14
\]

Would you like more details or further clarification? 

Here are some related questions:
1. How do you interpret the second derivative in real-world applications?
2. Can you find the second derivative of \( P = Q^4 + 5Q^3 + 3Q^2 + Q \)?
3. What does it mean if the second derivative is positive or negative?
4. How does the second derivative relate to concavity of a function?
5. Can you find the inflection point of a cubic function using the second derivative?

**Tip**: The second derivative is used to determine concavity and can indicate whether a function is concave up or concave down.

Which one of the following is the second derivatives of equation below?

P = 4Q^3 + 7Q^2 + 2Q

Discover how to find the second derivative of P = 4Q^3 + 7Q^2 + 2Q using calculus. This solution covers key differentiation concepts such as the power rule and the steps to compute first and second derivatives of polynomials. Suitable for students in Grades 10-12.

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