https://file.aiquickdraw.com/mathbot/file/1728166469659cd45veab.jpg

Let's solve the problems in the image using the General Power Rule for differentiation, which states:

\[
\frac{dy}{dx}(u^n) = n u^{n-1} \cdot \frac{du}{dx}
\]

### Problem 1: \( y = (2x^2 - 4)^5 \)

1. Here, \( u = 2x^2 - 4 \) and \( n = 5 \).
   - First, differentiate \( u = 2x^2 - 4 \):
     \[
     \frac{du}{dx} = \frac{d}{dx}(2x^2 - 4) = 4x
     \]
   - Now apply the power rule:
     \[
     \frac{dy}{dx} = 5(2x^2 - 4)^4 \cdot 4x
     \]
   - Simplifying:
     \[
     \frac{dy}{dx} = 20x(2x^2 - 4)^4
     \]

### Problem 2: \( y = (3x^2 + 1)^{\frac{7}{2}} \)

2. Here, \( u = 3x^2 + 1 \) and \( n = \frac{7}{2} \).
   - First, differentiate \( u = 3x^2 + 1 \):
     \[
     \frac{du}{dx} = \frac{d}{dx}(3x^2 + 1) = 6x
     \]
   - Now apply the power rule:
     \[
     \frac{dy}{dx} = \frac{7}{2}(3x^2 + 1)^{\frac{5}{2}} \cdot 6x
     \]
   - Simplifying:
     \[
     \frac{dy}{dx} = 21x(3x^2 + 1)^{\frac{5}{2}}
     \]

### Problem 3: \( y = (3x + 1)^2 (x^2 - 1)^{\frac{1}{2}} \)

3. This requires the product rule, combined with the power rule for each term.
   - Let \( u = (3x + 1)^2 \) and \( v = (x^2 - 1)^{\frac{1}{2}} \).

   First, differentiate \( u = (3x + 1)^2 \):
   \[
   \frac{du}{dx} = 2(3x + 1) \cdot 3 = 6(3x + 1)
   \]

   Now, differentiate \( v = (x^2 - 1)^{\frac{1}{2}} \):
   \[
   \frac{dv}{dx} = \frac{1}{2}(x^2 - 1)^{-\frac{1}{2}} \cdot 2x = \frac{x}{(x^2 - 1)^{\frac{1}{2}}}
   \]

   Using the product rule:
   \[
   \frac{dy}{dx} = u'v + uv'
   \]
   Substituting:
   \[
   \frac{dy}{dx} = 6(3x + 1)(x^2 - 1)^{\frac{1}{2}} + (3x + 1)^2 \cdot \frac{x}{(x^2 - 1)^{\frac{1}{2}}}
   \]

   This is the simplified derivative.

---

Let me know if you'd like further details or explanations on any part of the solution.

### Follow-up Questions:
1. How would you apply the chain rule in the case where the inner function is more complex?
2. What is the product rule and when do we use it in calculus?
3. Can the power rule be applied to negative exponents, and how?
4. What changes in the derivative if the exponent is a fraction less than 1?
5. How can you verify that the derivative obtained is correct?

**Tip**: For composite functions, always identify the inner and outer functions to apply the chain rule effectively.

Find the derivative of the following functions using the General Power formula: 1) y = (2x^2 - 4)^5, 2) y = (3x^2 + 1)^(7/2), 3) y = (3x + 1)^2 * (x^2 - 1)^(1/2)

Learn how to differentiate functions using the general power rule in calculus. This problem involves finding derivatives of composite functions and products, using the power rule and product rule. Suitable for students in advanced high school or early college calculus courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation