https://file.aiquickdraw.com/mathbot/file/17301599484141k9ojma2.jpg

To find the first and second derivatives, \( y' \) and \( y'' \), of the function \( y = x^2 \ln(9x) \), let’s analyze the work shown.

Given:
\[
y = x^2 \ln(9x)
\]

1. **Finding \( y' \):**
   Using the product rule \( (f \cdot g)' = f' g + f g' \) where \( f = x^2 \) and \( g = \ln(9x) \):
   
   - \( f = x^2 \Rightarrow f' = 2x \)
   - \( g = \ln(9x) \Rightarrow g' = \frac{1}{9x} \cdot 9 = \frac{1}{x} \)

   So,
   \[
   y' = f' g + f g' = 2x \ln(9x) + x^2 \cdot \frac{1}{x} = 2x \ln(9x) + \frac{x^2}{x} = 2x \ln(9x) + \frac{9x^2}{9x} = 2x \ln(9x)

Find y' and y'' for the function y = x^2 ln(9x).

Step-by-step solution to find the first and second derivatives, y' and y'', of the function y = x^2 ln(9x) using differentiation rules, including the product rule and logarithmic differentiation. Suitable for high school or introductory calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation