https://file.aiquickdraw.com/mathbot/file/1726801936166r2z10g7g.jpg

The problem in the image is asking to find the derivative of the function:

\[
h(x) = \frac{4}{x} - \frac{2}{x^3} + \frac{1}{x^4}
\]

We can rewrite the terms in the function in a form that makes differentiation easier:

\[
h(x) = 4x^{-1} - 2x^{-3} + x^{-4}
\]

Now, let's take the derivative of each term using the power rule, which states that \(\frac{d}{dx} [x^n] = nx^{n-1}\).

### Step-by-step solution:

1. The derivative of \(4x^{-1}\):
   \[
   \frac{d}{dx}[4x^{-1}] = 4 \cdot (-1)x^{-2} = -4x^{-2}
   \]

2. The derivative of \(-2x^{-3}\):
   \[
   \frac{d}{dx}[-2x^{-3}] = -2 \cdot (-3)x^{-4} = 6x^{-4}
   \]

3. The derivative of \(x^{-4}\):
   \[
   \frac{d}{dx}[x^{-4}] = (-4)x^{-5} = -4x^{-5}
   \]

### Putting it all together:

\[
h'(x) = -4x^{-2} + 6x^{-4} - 4x^{-5}
\]

This is the derivative of the given function.

Would you like more details on any step? Here are five related questions you might explore:

1. How can I apply the chain rule to more complex functions?
2. What is the power rule in differentiation?
3. How do we differentiate negative exponents?
4. Can we simplify the expression further after finding the derivative?
5. How do we compute second derivatives for this type of function?

**Tip:** Always rewrite fractions with exponents when differentiating—it simplifies the process!

Find the derivative of the function h(x) = 4/x - 2/x^3 + 1/x^4

This problem requires finding the derivative of the function h(x) = 4/x - 2/x^3 + 1/x^4 using the power rule. It covers key calculus concepts such as derivatives and negative exponents. 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