To find the derivative of the function \( f(x) = \sin(x) \cdot \tan(x) \cdot \ln(x) \) with respect to \( x \), we will use the **product rule** and the derivatives of each function involved. 

The product rule states that if \( u(x) \), \( v(x) \), and \( w(x) \) are functions of \( x \), then:

\[
\frac{d}{dx} [u(x) \cdot v(x) \cdot w(x)] = u'(x) \cdot v(x) \cdot w(x) + u(x) \cdot v'(x) \cdot w(x) + u(x) \cdot v(x) \cdot w'(x)
\]

### Step 1: Identify the functions
Let:
- \( u(x) = \sin(x) \)
- \( v(x) = \tan(x) \)
- \( w(x) = \ln(x) \)

### Step 2: Find the derivatives of each function
- \( u'(x) = \cos(x) \)
- \( v'(x) = \sec^2(x) \) (since \( \frac{d}{dx}[\tan(x)] = \sec^2(x) \))
- \( w'(x) = \frac{1}{x} \) (since \( \frac{d}{dx}[\ln(x)] = \frac{1}{x} \))

### Step 3: Apply the product rule
\[
\frac{d}{dx} [\sin(x) \cdot \tan(x) \cdot \ln(x)] = \cos(x) \cdot \tan(x) \cdot \ln(x) + \sin(x) \cdot \sec^2(x) \cdot \ln(x) + \sin(x) \cdot \tan(x) \cdot \frac{1}{x}
\]

### Final derivative:
\[
\frac{d}{dx} [\sin(x) \cdot \tan(x) \cdot \ln(x)] = \cos(x) \cdot \tan(x) \cdot \ln(x) + \sin(x) \cdot \sec^2(x) \cdot \ln(x) + \frac{\sin(x) \cdot \tan(x)}{x}
\]

Let me know if you want more details or clarifications!

### Related Questions:
1. How does the product rule apply when there are more than two functions?
2. What are the common derivatives of trigonometric functions?
3. How do we differentiate products of logarithmic and trigonometric functions?
4. What is the derivative of the function \( \tan(x) \)?
5. How do you handle composite functions when differentiating?

### Tip:
Always simplify the expression after using the product rule to make further calculations easier!

Learn how to differentiate the function sin(x)⋅tan(x)⋅ln(x) using the product rule. This step-by-step solution covers the key concepts of trigonometric and logarithmic functions, the product rule, and their respective derivatives. Ideal for students in advanced high school or undergraduate calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation