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Find​ (a) the slope of the curve at the given point​ P

To find the slope of the curve \( y = \sqrt{x} \) at the given point \( P(4, 2) \), we need to compute the derivative of \( y \) with respect to \( x \), which gives the slope of the tangent line at any point on the curve.

### Step-by-Step Solution:
1. The given equation is \( y = \sqrt{x} \).
   - Rewriting \( y = \sqrt{x} \) in a form that's easier to differentiate:
     \[
     y = x^{1/2}
     \]

2. Differentiate \( y = x^{1/2} \) with respect to \( x \):
   \[
   \frac{dy}{dx} = \frac{1}{2} x^{-1/2} = \frac{1}{2 \sqrt{x}}
   \]
   This gives the slope of the curve at any point \( x \).

3. Now, substitute \( x = 4 \) (the \( x \)-coordinate of point \( P(4, 2) \)) into the derivative to find the slope at this specific point:
   \[
   \frac{dy}{dx} \bigg|_{x = 4} = \frac{1}{2 \sqrt{4}} = \frac{1}{2 \times 2} = \frac{1}{4}
   \]

Thus, the slope of the curve at the point \( P(4, 2) \) is \( \frac{1}{4} \).

Would you like more details on the differentiation process or other aspects of this problem?

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Here are five related questions to deepen your understanding:
1. How would you find the equation of the tangent line to the curve at \( P(4, 2) \)?
2. What is the second derivative of \( y = \sqrt{x} \), and what does it tell us about the concavity of the curve?
3. How does the slope of the curve change as \( x \) increases?
4. Can you generalize the slope of any curve \( y = x^n \) at a given point?
5. What is the geometric meaning of the slope of the tangent line?

**Tip:** Always remember to rewrite radical expressions as exponents when differentiating, as it simplifies the process!

Find the slope of the curve at the given point P(4,2) for the equation y = sqrt(x).

Learn how to find the slope of the curve y = sqrt(x) at the point P(4,2) using differentiation. This problem covers important concepts like the power rule for differentiation and provides a step-by-step solution, suitable for students in Grades 10-12.

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