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 $(fg)' = f'g + fg'$.