Whats the derivative of h(x)= arctan(xsin(x))

To differentiate the function \( h(x) = \arctan(x \sin(x)) \), we need to apply both the chain rule and the product rule. Here's a step-by-step explanation:

### Step 1: Differentiate the outer function (arctan)
The derivative of \( \arctan(u) \) with respect to \( u \) is:
\[
\frac{d}{du}[\arctan(u)] = \frac{1}{1 + u^2}
\]
So, for \( h(x) = \arctan(x \sin(x)) \), the derivative will be:
\[
h'(x) = \frac{1}{1 + (x \sin(x))^2} \cdot \frac{d}{dx}[x \sin(x)]
\]

### Step 2: Differentiate the inner function \( x \sin(x) \)
Now, using the product rule to differentiate \( x \sin(x) \):
\[
\frac{d}{dx}[x \sin(x)] = \sin(x) + x \cos(x)
\]

### Step 3: Combine the results
Now, substitute this back into the derivative:
\[
h'(x) = \frac{\sin(x) + x \cos(x)}{1 + (x \sin(x))^2}
\]

So, the derivative of \( h(x) = \arctan(x \sin(x)) \) is:
\[
h'(x) = \frac{\sin(x) + x \cos(x)}{1 + (x \sin(x))^2}
\]

Would you like further details or clarification on this?

Here are 5 related questions for practice:
1. What is the derivative of \( h(x) = \arctan(2x) \)?
2. How would you differentiate \( h(x) = \arcsin(x \sin(x)) \)?
3. Can you differentiate \( h(x) = \arctan(\cos(x)) \)?
4. What is the second derivative of \( h(x) = \arctan(x^2) \)?
5. How would you find the derivative of \( h(x) = \arctan(e^x) \)?

**Tip:** When differentiating compositions like \( \arctan(f(x)) \), always remember to apply the chain rule carefully!

Whats the derivative of h(x)= arctan(xsin(x))?

Learn how to find the derivative of h(x) = arctan(xsin(x)) using the chain rule and product rule. This calculus problem involves trigonometric differentiation and is suitable for advanced high school or early college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation