If 

*f*(*x*) = (7*x*5 + 9*x*3 − 7)ln(*x*),

 find 

*f* *'*(*x*).

*f* *'*(*x*) =

To find the derivative \( f'(x) \) of the function

\[
f(x) = (7x^5 + 9x^3 - 7) \ln(x),
\]

we need to apply the **product rule** and the **chain rule** for differentiation.

### Step-by-step solution:

Let \( u(x) = 7x^5 + 9x^3 - 7 \) and \( v(x) = \ln(x) \).

#### 1. Apply the product rule:
\[
f'(x) = u'(x)v(x) + u(x)v'(x).
\]

#### 2. Differentiate \( u(x) \):
\[
u'(x) = \frac{d}{dx}(7x^5 + 9x^3 - 7) = 35x^4 + 27x^2.
\]

#### 3. Differentiate \( v(x) \):
\[
v'(x) = \frac{d}{dx}(\ln(x)) = \frac{1}{x}.
\]

#### 4. Substitute into the product rule:
\[
f'(x) = (35x^4 + 27x^2) \ln(x) + (7x^5 + 9x^3 - 7) \cdot \frac{1}{x}.
\]

#### 5. Simplify the second term:
\[
\frac{1}{x}(7x^5 + 9x^3 - 7) = 7x^4 + 9x^2 - \frac{7}{x}.
\]

Thus, the derivative is:

\[
f'(x) = (35x^4 + 27x^2) \ln(x) + 7x^4 + 9x^2 - \frac{7}{x}.
\]

This is the expression for \( f'(x) \).

---

Would you like further details or have any questions? Here are five related problems you might explore:

1. Differentiate \( f(x) = (5x^2 - 4x)\ln(x) \).
2. Find the second derivative of \( f(x) = (7x^5 + 9x^3 - 7)\ln(x) \).
3. Compute the derivative of \( f(x) = (2x^3 + x)\ln(x^2) \).
4. What is the derivative of \( f(x) = x^2 \ln(x^2 + 1) \)?
5. Find the derivative of \( f(x) = \ln(x^2 + 1) \).

**Tip**: The product rule is essential when differentiating functions that are products of two functions.

If f(x) = (7x^5 + 9x^3 − 7)ln(x), find f'(x).

Learn how to differentiate the function f(x) = (7x^5 + 9x^3 − 7)ln(x) using the product rule. This solution includes detailed steps for applying both the product rule and chain rule, suitable for Grades 11-12 calculus students.

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