Evaluate 𝑑𝑦𝑑𝑥(𝑥+1)(𝑥+ √𝑥) at x = 9

We are asked to evaluate the derivative \( \frac{dy}{dx} \) of the function \( y = (x+1)(x + \sqrt{x}) \) at \( x = 9 \).

### Step 1: Define the function
The function is given as:
\[
y = (x+1)(x + \sqrt{x})
\]

### Step 2: Use the product rule
We have a product of two functions, so we will apply the product rule, which states:
\[
\frac{d}{dx} [u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\]
Here, let:
- \( u(x) = x + 1 \)
- \( v(x) = x + \sqrt{x} \)

### Step 3: Find the derivatives of \( u(x) \) and \( v(x) \)
1. \( u(x) = x + 1 \), so \( u'(x) = 1 \).
2. \( v(x) = x + \sqrt{x} \), so we need to find \( v'(x) \).

To differentiate \( v(x) \):
\[
v(x) = x + x^{1/2}
\]
The derivative is:
\[
v'(x) = 1 + \frac{1}{2}x^{-1/2} = 1 + \frac{1}{2\sqrt{x}}
\]

### Step 4: Apply the product rule
Now apply the product rule:
\[
\frac{dy}{dx} = u'(x)v(x) + u(x)v'(x)
\]
Substitute \( u(x) \), \( v(x) \), and their derivatives:
\[
\frac{dy}{dx} = (1)(x + \sqrt{x}) + (x+1)\left(1 + \frac{1}{2\sqrt{x}}\right)
\]
Simplify:
\[
\frac{dy}{dx} = x + \sqrt{x} + (x+1)\left(1 + \frac{1}{2\sqrt{x}}\right)
\]

### Step 5: Evaluate at \( x = 9 \)
Now substitute \( x = 9 \) into the expression:
\[
\frac{dy}{dx} = 9 + \sqrt{9} + (9+1)\left(1 + \frac{1}{2\sqrt{9}}\right)
\]
Simplify:
\[
\frac{dy}{dx} = 9 + 3 + 10\left(1 + \frac{1}{6}\right)
\]
\[
\frac{dy}{dx} = 12 + 10\left(\frac{7}{6}\right)
\]
\[
\frac{dy}{dx} = 12 + \frac{70}{6}
\]
\[
\frac{dy}{dx} = 12 + 11.67
\]
\[
\frac{dy}{dx} \approx 23.67
\]

So, the value of \( \frac{dy}{dx} \) at \( x = 9 \) is approximately **23.67**.

Would you like any additional explanation or details on this?

### Related Questions:
1. What is the general formula for the product rule in calculus?
2. How do we differentiate functions with square roots?
3. What are other applications of the product rule in physics?
4. How does the chain rule relate to differentiating nested functions?
5. What happens if you differentiate a product of three functions?

**Tip**: When differentiating functions involving radicals like \( \sqrt{x} \), express them as exponents (e.g., \( x^{1/2} \)) to make the differentiation easier.

Evaluate 𝑑𝑦/𝑑𝑥 (𝑥+1)(𝑥+ √𝑥) at x = 9

Explore how to differentiate the function (x+1)(x + √x) using the product rule and evaluate the derivative at x = 9. The problem applies key calculus concepts such as the product rule, square roots, and differentiation. Suitable for advanced high school or early college students.

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