evaluate f(4) where f(x) =x$$ \sqrt{2x+1} $$ by differentiating

To evaluate $ f(4) $ for the function $ f(x) = x\sqrt{2x + 1} $ by differentiating, we need to follow these steps:

1. **Differentiate** the function $ f(x) $ with respect to $ x $ using the product rule.
2. **Evaluate** the derivative at $ x = 4 $.

### Step 1: Differentiating $ f(x) = x \sqrt{2x + 1} $

The function is a product of two parts: 
- $ x $, and 
- $ \sqrt{2x + 1} $.

Using the **product rule**: 
\[
(fg)' = f'g + fg'
\]

Here, let $ f(x) = x $ and $ g(x) = \sqrt{2x + 1} $.

- The derivative of $ x $ is:
  \[
  \frac{d}{dx} [x] = 1
  \]
- The derivative of $ \sqrt{2x + 1} $ is found using the **chain rule**:
  \[
  \frac{d}{dx} \left[ \sqrt{2x + 1} \right] = \frac{1}{2\sqrt{2x + 1}} \cdot \frac{d}{dx} [2x + 1] = \frac{1}{2\sqrt{2x + 1}} \cdot 2 = \frac{1}{\sqrt{2x + 1}}
  \]

Thus, the derivative of $ f(x) = x \sqrt{2x + 1} $ is:
\[
f'(x) = 1 \cdot \sqrt{2x + 1} + x \cdot \frac{1}{\sqrt{2x + 1}}
\]
Simplifying:
\[
f'(x) = \frac{2x + 1 + x}{\sqrt{2x + 1}} = \frac{3x + 1}{\sqrt{2x + 1}}
\]

### Step 2: Evaluating at $ x = 4 $

Now substitute $ x = 4 $ into the derivative:
\[
f'(4) = \frac{3(4) + 1}{\sqrt{2(4) + 1}} = \frac{12 + 1}{\sqrt{8 + 1}} = \frac{13}{\sqrt{9}} = \frac{13}{3}
\]

Thus, the value of the derivative at $ x = 4 $ is:
\[
f'(4) = \frac{13}{3}
\]

### Conclusion:
By differentiating, we find that $ f'(4) = \frac{13}{3} $.

Would you like more details or explanations on any step?

### Related Questions:
1. What is the general rule for differentiating square root functions?
2. How is the product rule applied in more complex cases?
3. Can the chain rule be used directly for a composite function like $ \sqrt{2x+1} $?
4. What is the meaning of evaluating the derivative at a point?
5. How do we interpret the result of the derivative in practical applications?

**Tip:** When applying the chain rule, always differentiate the inner function first before multiplying by the derivative of the outer function.

evaluate f(4) where f(x) = x√(2x+1) by differentiating

Learn how to differentiate the function f(x) = x√(2x+1) using the product and chain rules, and evaluate the derivative at x = 4. This step-by-step guide covers key calculus concepts such as the product rule and chain rule, suitable for students in Grades 11-12 or college-level calculus.

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