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

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

### Step-by-step process:

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

We recognize that this is the product of two functions:
- \( u(x) = x^2 \)
- \( v(x) = \ln(9x) \)

### 1. Differentiate using the product rule:
The product rule states:
\[
(fg)' = f'g + fg'
\]
So:
\[
f'(x) = u'(x) v(x) + u(x) v'(x)
\]

### 2. Differentiate \( u(x) = x^2 \):
\[
u'(x) = 2x
\]

### 3. Differentiate \( v(x) = \ln(9x) \):
To differentiate \( \ln(9x) \), use the chain rule:
\[
v'(x) = \frac{1}{9x} \cdot 9 = \frac{1}{x}
\]

### 4. Apply the product rule:
\[
f'(x) = (2x) \ln(9x) + x^2 \cdot \frac{1}{x}
\]

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

Thus, the derivative of \( f(x) = x^2 \ln(9x) \) is:
\[
f'(x) = 2x \ln(9x) + x
\]

Would you like more details or have any questions? 

Here are 5 related questions for practice:

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

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

For the following exercises, find f’(x) for each function. f(x)= x^2 ln 9x

Learn how to find the derivative of the function f(x) = x^2 ln(9x) using the product rule and chain rule. This step-by-step solution covers key calculus concepts such as differentiation, logarithmic functions, and rule application. Suitable for Grades 11-12 and early college students.

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