https://r2.erweima.ai/mathbot/image/17263091331244icybhvk.jpg

It seems that the image contains a list of functions for which you are asked to calculate derivatives. From what I can see:

- \( f(x) = 5\sqrt{x} \)
- \( g(x) = \frac{-3}{x} \)
- \( h(x) = 5x^5 - 2x^4 + x^2 - 2 \)

I'll go ahead and compute the derivatives for each of these functions.

### 1. \( f(x) = 5\sqrt{x} \)

We can rewrite \( \sqrt{x} \) as \( x^{1/2} \).

\[
f(x) = 5x^{1/2}
\]

Now, differentiate:

\[
f'(x) = 5 \cdot \frac{1}{2} x^{-1/2} = \frac{5}{2\sqrt{x}}
\]

### 2. \( g(x) = \frac{-3}{x} \)

We can rewrite this as \( g(x) = -3x^{-1} \).

Now, differentiate:

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

### 3. \( h(x) = 5x^5 - 2x^4 + x^2 - 2 \)

To differentiate this polynomial:

\[
h'(x) = 5 \cdot 5x^4 - 2 \cdot 4x^3 + 2x = 25x^4 - 8x^3 + 2x
\]

### Summary of Derivatives:
1. \( f'(x) = \frac{5}{2\sqrt{x}} \)
2. \( g'(x) = \frac{3}{x^2} \)
3. \( h'(x) = 25x^4 - 8x^3 + 2x \)

Let me know if you need any further details or clarification!

---

Here are 5 related questions for further exploration:
1. How does the power rule for differentiation apply to fractional exponents?
2. What is the derivative of the general power function \( x^n \)?
3. How can you compute the derivative of a function involving a negative exponent?
4. What is the derivative of \( \sqrt{x} \) using the chain rule?
5. How do you differentiate a constant multiplied by a function?

**Tip**: For square roots and other fractional powers, rewrite them as exponents to easily apply the power rule!

Calculer les dérivées des fonctions suivantes: f(x) = 5√x, g(x) = -3/x, h(x) = 5x^5 - 2x^4 + x^2 - 2.

This lesson covers how to calculate derivatives for the functions f(x) = 5√x, g(x) = -3/x, and h(x) = 5x^5 - 2x^4 + x^2 - 2. Using the power rule and differentiation techniques for square roots and rational functions, you'll learn how to find the derivatives step by step.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation