https://file.aiquickdraw.com/mathbot/file/1729368479150nwfqt9l9.jpg

We are tasked with finding the derivative of the given function \( f(x) = \left( \arctan(x) \right)^4 + 3\arctan(x) + 5 \).

### Step 1: Differentiate each term individually.

1. **Derivative of \( \left( \arctan(x) \right)^4 \):**
   - Using the chain rule, we get:
     \[
     \frac{d}{dx} \left( \arctan(x) \right)^4 = 4 \left( \arctan(x) \right)^3 \cdot \frac{d}{dx} \arctan(x).
     \]
     We know that \( \frac{d}{dx} \arctan(x) = \frac{1}{1+x^2} \), so:
     \[
     \frac{d}{dx} \left( \arctan(x) \right)^4 = 4 \left( \arctan(x) \right)^3 \cdot \frac{1}{1+x^2}.
     \]

2. **Derivative of \( 3\arctan(x) \):**
   - This is straightforward. Using the derivative of \( \arctan(x) \), we get:
     \[
     \frac{d}{dx} 3\arctan(x) = 3 \cdot \frac{1}{1+x^2}.
     \]

3. **Derivative of the constant 5:**
   - The derivative of a constant is 0.

### Step 2: Combine all the terms.
\[
f'(x) = 4 \left( \arctan(x) \right)^3 \cdot \frac{1}{1+x^2} + 3 \cdot \frac{1}{1+x^2}.
\]

So, the derivative of the function is:
\[
f'(x) = \frac{4 \left( \arctan(x) \right)^3}{1+x^2} + \frac{3}{1+x^2}.
\]

Would you like further explanation on any of the steps, or do you have any specific questions?

### Related Questions:
1. What is the derivative of \( \arctan(x)^n \) for any power \( n \)?
2. How does the chain rule apply in cases like \( \left( \arctan(x) \right)^4 \)?
3. What is the geometric interpretation of the arctangent function's derivative?
4. Can the function \( f(x) = \arctan(x) \) be generalized for multivariable calculus?
5. How can this derivative be used in solving integral problems involving \( \arctan(x) \)?

### Tip:
When differentiating powers of functions, always use the chain rule to account for the inner function's derivative!

What is the derivative of f(x) = (arctan(x))^4 + 3arctan(x) + 5?

Learn how to find the derivative of the function f(x) = (arctan(x))^4 + 3arctan(x) + 5 using differentiation rules like the chain rule. Step-by-step solution provided for calculus students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation