Double derivatives of (x^2-1)^2 with respect to x

To find the double derivative of \( (x^2 - 1)^2 \) with respect to \( x \), we follow these steps:

### Step 1: First Derivative
First, differentiate \( (x^2 - 1)^2 \) with respect to \( x \).

Let \( u = x^2 - 1 \). Then,

\[
\frac{d}{dx} \left( u^2 \right) = 2u \cdot \frac{du}{dx}
\]

Since \( u = x^2 - 1 \), we have:

\[
\frac{du}{dx} = 2x
\]

Therefore, the first derivative is:

\[
\frac{d}{dx} \left( (x^2 - 1)^2 \right) = 2(x^2 - 1) \cdot 2x = 4x(x^2 - 1)
\]

### Step 2: Second Derivative
Next, differentiate the result \( 4x(x^2 - 1) \) with respect to \( x \) to find the second derivative.

Apply the product rule:

\[
\frac{d}{dx} \left( 4x(x^2 - 1) \right) = 4 \cdot \frac{d}{dx}(x) \cdot (x^2 - 1) + 4x \cdot \frac{d}{dx}(x^2 - 1)
\]

First term:

\[
4 \cdot 1 \cdot (x^2 - 1) = 4(x^2 - 1)
\]

Second term:

\[
4x \cdot 2x = 8x^2
\]

So, the second derivative is:

\[
\frac{d^2}{dx^2} \left( (x^2 - 1)^2 \right) = 4(x^2 - 1) + 8x^2
\]

Simplifying this:

\[
\frac{d^2}{dx^2} \left( (x^2 - 1)^2 \right) = 4x^2 - 4 + 8x^2 = 12x^2 - 4
\]

Thus, the double derivative of \( (x^2 - 1)^2 \) with respect to \( x \) is \( 12x^2 - 4 \).

---

Would you like further details or have any other questions?

Here are 5 related questions for further exploration:

1. How would you find the third derivative of \( (x^2 - 1)^2 \)?
2. How does the second derivative relate to the concavity of the function \( (x^2 - 1)^2 \)?
3. What are the critical points of the function \( (x^2 - 1)^2 \)?
4. How would you apply the chain rule to find derivatives of more complex functions?
5. What is the geometric interpretation of the first and second derivatives?

**Tip:** The second derivative gives important information about the concavity of a function and can help in determining points of inflection.

Learn how to find the double derivatives of (x^2 - 1)^2 with detailed step-by-step solutions. This problem covers differentiation of polynomials, application of the product rule, and is suitable for college-level mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation