https://file.aiquickdraw.com/mathbot/file/1726720466311ak7chiwj.jpg

We are tasked with finding \( f'(4) \) for the function \( f(x) = 6\sqrt{x} - \frac{2}{\sqrt{x}} \).

### Step 1: Simplify the function
Let's rewrite \( f(x) \) in terms of powers of \( x \):
\[
f(x) = 6x^{1/2} - 2x^{-1/2}
\]

### Step 2: Differentiate the function
To find the derivative \( f'(x) \), we apply the power rule:
- The derivative of \( 6x^{1/2} \) is \( \frac{6}{2}x^{-1/2} = 3x^{-1/2} \)
- The derivative of \( -2x^{-1/2} \) is \( \frac{-2}{2}x^{-3/2} = -x^{-3/2} \)

So, the derivative is:
\[
f'(x) = 3x^{-1/2} - x^{-3/2}
\]

### Step 3: Substitute \( x = 4 \) to find \( f'(4) \)
Now, substitute \( x = 4 \) into the derivative:
\[
f'(4) = 3(4^{-1/2}) - (4^{-3/2})
\]
We know:
\[
4^{-1/2} = \frac{1}{2}, \quad 4^{-3/2} = \frac{1}{8}
\]
Substitute these values:
\[
f'(4) = 3 \times \frac{1}{2} - \frac{1}{8} = \frac{3}{2} - \frac{1}{8}
\]

### Step 4: Simplify the result
To subtract these fractions, find a common denominator:
\[
\frac{3}{2} = \frac{12}{8}
\]
Thus:
\[
f'(4) = \frac{12}{8} - \frac{1}{8} = \frac{11}{8}
\]

So, \( f'(4) = \frac{11}{8} \).

Let me know if you want further clarifications, and here are some related questions you might want to explore:

1. How do you apply the chain rule when differentiating more complex functions?
2. What happens if the function includes logarithmic or exponential terms?
3. Can you solve for the second derivative of this function and interpret its meaning?
4. How do you approach solving limits involving square roots?
5. What role do fractional exponents play in simplifying derivatives?

**Tip**: Always simplify a function into powers of \( x \) to make differentiation easier, especially when dealing with square roots.

Given the function f(x) = 6√x − 2/√x, find f'(4). Express your answer as a single fraction in simplest form.

Learn how to differentiate the function f(x) = 6√x − 2/√x and find f'(4). This problem involves applying the power rule for derivatives and simplifying fractional exponents. A step-by-step solution is provided for high school calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation