Math Problem Statement

For the following exercises, find f’(x) for each function. f(x)= x2 ln 9x

Solution

To find the derivative f(x)f'(x) for the function f(x)=x2ln(9x)f(x) = x^2 \ln(9x), we can use the product rule and chain rule.

Step-by-step process:

Given: f(x)=x2ln(9x)f(x) = x^2 \ln(9x)

We recognize that this is the product of two functions:

  • u(x)=x2u(x) = x^2
  • v(x)=ln(9x)v(x) = \ln(9x)

1. Differentiate using the product rule:

The product rule states: (fg)=fg+fg(fg)' = f'g + fg' So: f(x)=u(x)v(x)+u(x)v(x)f'(x) = u'(x) v(x) + u(x) v'(x)

2. Differentiate u(x)=x2u(x) = x^2:

u(x)=2xu'(x) = 2x

3. Differentiate v(x)=ln(9x)v(x) = \ln(9x):

To differentiate ln(9x)\ln(9x), use the chain rule: v(x)=19x9=1xv'(x) = \frac{1}{9x} \cdot 9 = \frac{1}{x}

4. Apply the product rule:

f(x)=(2x)ln(9x)+x21xf'(x) = (2x) \ln(9x) + x^2 \cdot \frac{1}{x}

Simplify the second term: f(x)=2xln(9x)+xf'(x) = 2x \ln(9x) + x

Thus, the derivative of f(x)=x2ln(9x)f(x) = x^2 \ln(9x) is: f(x)=2xln(9x)+xf'(x) = 2x \ln(9x) + x

Would you like more details or have any questions?

Here are 5 related questions for practice:

  1. Differentiate f(x)=x3ln(4x)f(x) = x^3 \ln(4x).
  2. Find f(x)f'(x) for f(x)=x2ln(x2)f(x) = x^2 \ln(x^2).
  3. Differentiate f(x)=x2ln(5x3)f(x) = x^2 \ln(5x^3).
  4. What is the derivative of f(x)=xln(3x2)f(x) = x \ln(3x^2)?
  5. Find the derivative of f(x)=x4ln(x)f(x) = x^4 \ln(x).

Tip: When differentiating products of functions, always remember the product rule formula (fg)=fg+fg(fg)' = f'g + fg'.

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Math Problem Analysis

Mathematical Concepts

Differentiation
Product Rule
Chain Rule
Logarithmic Functions

Formulas

Product Rule: (fg)' = f'g + fg'
Chain Rule: (ln(9x))' = 1/x

Theorems

Product Rule
Chain Rule

Suitable Grade Level

Grades 11-12, Early College