https://file.aiquickdraw.com/mathbot/file/1727567031680maoorf0e.jpg

We are given the function:

\[
h(x) = \frac{1}{x^2} + \frac{1}{x} + x
\]

Now, we need to compute its derivative \( h'(x) \).

### Step-by-step solution:

1. **Differentiate \( \frac{1}{x^2} \):**
   Using the power rule and chain rule, we treat \( \frac{1}{x^2} \) as \( x^{-2} \).
   \[
   \frac{d}{dx}\left( x^{-2} \right) = -2x^{-3} = -\frac{2}{x^3}
   \]

2. **Differentiate \( \frac{1}{x} \):**
   Again, using the power rule, \( \frac{1}{x} \) is \( x^{-1} \).
   \[
   \frac{d}{dx}\left( x^{-1} \right) = -x^{-2} = -\frac{1}{x^2}
   \]

3. **Differentiate \( x \):**
   The derivative of \( x \) is simply 1.

### Final derivative:
Combining all these results together, we get:

\[
h'(x) = -\frac{2}{x^3} - \frac{1}{x^2} + 1
\]

This is the derivative of the given function \( h(x) \).

Would you like more details on any specific part? 

Here are some related questions you might want to consider:
1. How do you apply the power rule to negative exponents?
2. What is the chain rule, and how is it used in differentiation?
3. How do you differentiate products or quotients of functions?
4. Can you explain implicit differentiation for more complex functions?
5. How do you apply the second derivative to analyze the concavity of a function?

**Tip:** When differentiating terms with powers of \( x \), always rewrite the expression as a power of \( x \) (e.g., \( \frac{1}{x^n} = x^{-n} \)) before applying the power rule.

Learn how to find the derivative of the function h(x) = 1/x^2 + 1/x + x using the power rule and chain rule. This calculus problem involves key differentiation concepts and is suitable for 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