For the function y=(4x+3)(2x^2+7x−1), find d^2y/dx^2


To find the second derivative of \( y = (4x + 3)(2x^2 + 7x - 1) \), we will proceed step by step.

### Step 1: Expand the function

First, expand the expression for \( y \):

\[
y = (4x + 3)(2x^2 + 7x - 1)
\]

Distribute the terms:

\[
y = 4x(2x^2 + 7x - 1) + 3(2x^2 + 7x - 1)
\]

Now, multiply each term:

\[
y = 4x(2x^2) + 4x(7x) + 4x(-1) + 3(2x^2) + 3(7x) + 3(-1)
\]

\[
y = 8x^3 + 28x^2 - 4x + 6x^2 + 21x - 3
\]

Simplify the expression:

\[
y = 8x^3 + (28x^2 + 6x^2) + (-4x + 21x) - 3
\]

\[
y = 8x^3 + 34x^2 + 17x - 3
\]

### Step 2: Find the first derivative \( \frac{dy}{dx} \)

Now differentiate \( y = 8x^3 + 34x^2 + 17x - 3 \) with respect to \( x \):

\[
\frac{dy}{dx} = \frac{d}{dx}(8x^3) + \frac{d}{dx}(34x^2) + \frac{d}{dx}(17x) + \frac{d}{dx}(-3)
\]

Apply the power rule to each term:

\[
\frac{dy}{dx} = 24x^2 + 68x + 17
\]

### Step 3: Find the second derivative \( \frac{d^2y}{dx^2} \)

Differentiate \( \frac{dy}{dx} = 24x^2 + 68x + 17 \) again with respect to \( x \):

\[
\frac{d^2y}{dx^2} = \frac{d}{dx}(24x^2) + \frac{d}{dx}(68x) + \frac{d}{dx}(17)
\]

Apply the power rule again:

\[
\frac{d^2y}{dx^2} = 48x + 68
\]

### Final Answer:
\[
\frac{d^2y}{dx^2} = 48x + 68
\]

Would you like more details on any step, or do you have further questions?

Here are 5 related questions for practice:
1. What is the first derivative of \( y = (5x + 2)(3x^2 + 4x - 1) \)?
2. How do you find the second derivative of a product of two functions using the product rule?
3. How does the chain rule apply in finding higher-order derivatives?
4. What is the second derivative of \( y = x^3 + 5x^2 - 4x + 1 \)?
5. How can the second derivative help determine concavity and points of inflection of a curve?

**Tip:** When taking derivatives of a product, expanding first simplifies the process but you can also use the product rule directly.

For the function y=(4x+3)(2x^2+7x−1), find d^2y/dx^2

Learn how to find the second derivative of the function y = (4x + 3)(2x^2 + 7x − 1) with a detailed step-by-step solution. This problem covers polynomial expansion, first and second derivatives using the power and product rules. Ideal for students in Grades 11-12 or early college.

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