https://file.aiquickdraw.com/mathbot/file/1726788627287q67cmnkc.jpg

We are given the function \( f(x) = \frac{4x^2 + 8x + 7}{\sqrt{x}} \), and we need to find its derivative \( f'(x) \).

### Step 1: Rewrite the function
First, express \( \frac{1}{\sqrt{x}} \) as \( x^{-\frac{1}{2}} \). This makes the function easier to differentiate.

\[
f(x) = (4x^2 + 8x + 7) \cdot x^{-\frac{1}{2}}
\]

### Step 2: Apply the product rule
We now differentiate using the product rule. The product rule states that for two functions \( u(x) \) and \( v(x) \), the derivative is:

\[
(uv)' = u'v + uv'
\]

Let:
- \( u(x) = 4x^2 + 8x + 7 \)
- \( v(x) = x^{-\frac{1}{2}} \)

### Step 3: Differentiate \( u(x) \) and \( v(x) \)

- \( u'(x) = \frac{d}{dx}(4x^2 + 8x + 7) = 8x + 8 \)
- \( v'(x) = \frac{d}{dx}(x^{-\frac{1}{2}}) = -\frac{1}{2} x^{-\frac{3}{2}} \)

### Step 4: Apply the product rule

Now, use the product rule to find \( f'(x) \):

\[
f'(x) = (8x + 8) \cdot x^{-\frac{1}{2}} + (4x^2 + 8x + 7) \cdot \left(-\frac{1}{2} x^{-\frac{3}{2}}\right)
\]

### Step 5: Simplify the expression

Now, simplify each term:
- First term: \( (8x + 8) \cdot x^{-\frac{1}{2}} = 8x^{\frac{1}{2}} + 8x^{-\frac{1}{2}} \)
- Second term: \( (4x^2 + 8x + 7) \cdot \left(-\frac{1}{2} x^{-\frac{3}{2}}\right) = -2x^{\frac{1}{2}} - 4x^{-\frac{1}{2}} - \frac{7}{2}x^{-\frac{3}{2}} \)

Now combine the like terms:

\[
f'(x) = (8x^{\frac{1}{2}} - 2x^{\frac{1}{2}}) + (8x^{-\frac{1}{2}} - 4x^{-\frac{1}{2}}) - \frac{7}{2}x^{-\frac{3}{2}}
\]

\[
f'(x) = 6x^{\frac{1}{2}} + 4x^{-\frac{1}{2}} - \frac{7}{2}x^{-\frac{3}{2}}
\]

This is the derivative of the given function.

### Final Answer:
\[
f'(x) = 6x^{\frac{1}{2}} + 4x^{-\frac{1}{2}} - \frac{7}{2}x^{-\frac{3}{2}}
\]

Would you like a more detailed explanation of any step or have further questions?

Here are 5 related questions for practice:
1. What is the derivative of \( \frac{x^3 + 2x}{\sqrt{x}} \)?
2. How does the product rule apply when one of the functions is a constant?
3. Can you find the second derivative of \( f(x) \)?
4. What is the derivative of \( \frac{5x^2 + 3x + 1}{x} \)?
5. How do you simplify expressions involving negative exponents?

**Tip:** When differentiating a quotient, it's often helpful to rewrite it as a product using negative exponents.

If f(x) = (4x^2 + 8x + 7) / sqrt(x), find f'(x).

Learn how to differentiate the function f(x) = (4x^2 + 8x + 7) / sqrt(x) using the product rule and exponent rules. This step-by-step solution simplifies the expression and combines like terms. Suitable for students in Grades 11-12 and early college courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation