https://file.aiquickdraw.com/mathbot/file/17300692357384rgqbju8.jpg

To find the derivative \( y' \) of the function \( y = (6x + 1)^5 (x^4 - 6)^6 \) using logarithmic differentiation, follow these steps:

1. **Take the natural logarithm of both sides**:
   \[
   \ln y = \ln \left( (6x + 1)^5 (x^4 - 6)^6 \right)
   \]

2. **Use the properties of logarithms** (specifically, \(\ln(ab) = \ln a + \ln b\) and \(\ln(a^n) = n \ln a\)):
   \[
   \ln y = 5 \ln(6x + 1) + 6 \ln(x^4 - 6)
   \]

3. **Differentiate both sides with respect to \( x \)**:
   Using implicit differentiation on the left side, we get:
   \[
   \frac{1}{y} \cdot y' = 5 \cdot \frac{1}{6x + 1} \cdot 6 + 6 \cdot \frac{1}{x^4 - 6} \cdot 4x^3
   \]

4. **Simplify the expression**:
   \[
   \frac{y'}{y} = \frac{30}{6x + 1} + \frac{24x^3}{x^4 - 6}
   \]

5. **Solve for \( y' \) by multiplying both sides by \( y \)**:
   Substitute back the expression for \( y = (6x + 1)^5 (x^4 - 6)^6 \):
   \[
   y' = \left( \frac{30}{6x + 1} + \frac{24x^3}{x^4 - 6} \right) (6x + 1)^5 (x^4 - 6)^6
   \]

So, the derivative \( y' \) is:
\[
y' = \left( \frac{30}{6x + 1} + \frac{24x^3}{x^4 - 6} \right) (6x + 1)^5 (x^4 - 6)^6
\]

Would you like more details on any of these steps, or do you have any questions?

---

### Related Questions
1. How does logarithmic differentiation help with complex products and quotients?
2. What is the significance of using implicit differentiation in logarithmic differentiation?
3. How would the steps change if there were more terms in the function \( y \)?
4. Could we solve this problem without logarithmic differentiation? What are the benefits of using it here?
5. How does the chain rule apply in logarithmic differentiation?

### Tip
Using logarithmic differentiation is especially useful for functions that are products or quotients of several terms with exponents. It simplifies differentiation by turning products into sums, making the process easier to manage.

Use logarithmic differentiation to find the derivative of the function y = (6x + 1)^5 (x^4 - 6)^6.

Find the derivative of the function y = (6x + 1)^5 (x^4 - 6)^6 using logarithmic differentiation. This problem covers key concepts in calculus, including implicit differentiation and properties of logarithms, making it suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation