https://file.aiquickdraw.com/mathbot/file/1729910620539e4qu0u3o.jpg

To solve this problem, we will use **logarithmic differentiation**.

Given:
\[
y = (4x + 1)^5 (x^4 - 1)^6
\]

### Step 1: Take the natural logarithm of both sides
Take the natural logarithm on both sides to simplify the differentiation of the product:
\[
\ln y = \ln\left((4x + 1)^5 (x^4 - 1)^6\right)
\]

Using the property \(\ln(ab) = \ln a + \ln b\), we get:
\[
\ln y = \ln\left((4x + 1)^5\right) + \ln\left((x^4 - 1)^6\right)
\]

Now, apply the property \(\ln(a^b) = b \ln a\):
\[
\ln y = 5 \ln(4x + 1) + 6 \ln(x^4 - 1)
\]

### Step 2: Differentiate both sides with respect to \(x\)
Now, differentiate both sides with respect to \(x\). Remember, the derivative of \(\ln y\) with respect to \(x\) is \(\frac{1}{y} \cdot \frac{dy}{dx} = \frac{y'}{y}\).

\[
\frac{y'}{y} = 5 \cdot \frac{1}{4x + 1} \cdot 4 + 6 \cdot \frac{1}{x^4 - 1} \cdot 4x^3
\]

Simplifying each term:

1. For the first term:
   \[
   5 \cdot \frac{4}{4x + 1} = \frac{20}{4x + 1}
   \]

2. For the second term:
   \[
   6 \cdot \frac{4x^3}{x^4 - 1} = \frac{24x^3}{x^4 - 1}
   \]

So we get:
\[
\frac{y'}{y} = \frac{20}{4x + 1} + \frac{24x^3}{x^4 - 1}
\]

### Step 3: Solve for \(y'\)
Multiply both sides by \(y\) to isolate \(y'\):
\[
y' = y \left( \frac{20}{4x + 1} + \frac{24x^3}{x^4 - 1} \right)
\]

Since \(y = (4x + 1)^5 (x^4 - 1)^6\), substitute this back in:
\[
y' = (4x + 1)^5 (x^4 - 1)^6 \left( \frac{20}{4x + 1} + \frac{24x^3}{x^4 - 1} \right)
\]

This is the derivative \(y'\) using logarithmic differentiation.

Would you like a deeper explanation or have any further questions?

---

Here are five related questions for practice:
1. How would you approach the derivative if the exponents were negative?
2. Can you differentiate \(y = (2x + 3)^3(x^2 - 4)^4\) using logarithmic differentiation?
3. How would you apply logarithmic differentiation to a function with three terms multiplied together?
4. What is the benefit of logarithmic differentiation over other methods?
5. Can logarithmic differentiation be used on functions that contain trigonometric terms?

**Tip:** Logarithmic differentiation is especially useful when dealing with products or quotients of terms with powers, as it turns multiplication into addition, simplifying the process.

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

Learn how to use logarithmic differentiation to find the derivative of the function y = (4x + 1)^5 (x^4 - 1)^6. This problem covers key calculus concepts, including the product rule and chain rule, making it suitable for high school 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