Differentiat with respect to x with nth deg

To differentiate the expression \((x^2 - 1)^n\) with respect to \(x\) and consider the derivative with respect to the general \(n\)-th degree, we use the **chain rule**.

### Given:
\[
y = (x^2 - 1)^n
\]

### First Derivative with Respect to \(x\)
To differentiate this with respect to \(x\):

1. **Apply the chain rule**: The chain rule states that if you have a composite function \( y = f(g(x)) \), then its derivative is given by:
   \[
   \frac{dy}{dx} = \frac{df}{dg} \cdot \frac{dg}{dx}
   \]
   
2. **Differentiate the outer function** \(f(g(x)) = g(x)^n\) where \(g(x) = x^2 - 1\):
   \[
   \frac{dy}{dg} = n \cdot (g(x))^{n-1} = n \cdot (x^2 - 1)^{n-1}
   \]
   
3. **Differentiate the inner function** \(g(x) = x^2 - 1\):
   \[
   \frac{dg}{dx} = \frac{d}{dx}(x^2 - 1) = 2x
   \]
   
4. **Multiply them together**:
   \[
   \frac{dy}{dx} = n \cdot (x^2 - 1)^{n-1} \cdot 2x
   \]

### Result:
\[
\frac{dy}{dx} = 2n \cdot x \cdot (x^2 - 1)^{n-1}
\]

This expression represents the derivative of \((x^2 - 1)^n\) with respect to \(x\).

### Higher-Order Derivatives
If you are interested in the \(k\)-th derivative (where \(k\) is some integer), you'd continue differentiating the result above multiple times. The process becomes more complex as you increase the order of the derivative, involving applying the product and chain rules repeatedly.

Would you like to explore higher-order derivatives, or do you have any further questions?

### Related Questions:
1. How do you find the second derivative of \((x^2 - 1)^n\) with respect to \(x\)?
2. What is the derivative of \((x^2 - 1)^n\) with respect to \(x\) for specific values of \(n\)?
3. How does the derivative change if you have a different exponent, such as \((x^2 - 1)^{n/2}\)?
4. Can you generalize the derivative \(\frac{dy}{dx}\) for any function of the form \((g(x))^n\)?
5. What is the application of the derivative \(\frac{dy}{dx}\) in physics or engineering problems?

### Tip:
When differentiating functions involving exponents, always remember to apply the chain rule carefully, as missing it can lead to incorrect results.

Learn how to differentiate the expression (x^2 - 1)^n with respect to x using the chain rule. Explore step-by-step solutions and applications of differentiation in algebraic contexts. Suitable for advanced high school and college students.

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