To find the derivative of the function \( y = \sqrt{5x + 2} \), we can apply the **chain rule**. Here's how:

### Step-by-step solution:

1. Rewrite the function in exponent form:
   \[
   y = (5x + 2)^{1/2}
   \]
   
2. Use the chain rule, which states that if \( y = [u(x)]^n \), then:
   \[
   \frac{dy}{dx} = n \cdot [u(x)]^{n-1} \cdot \frac{du}{dx}
   \]
   Here, \( u(x) = 5x + 2 \) and \( n = \frac{1}{2} \).

3. Differentiate:
   - First, apply the power rule: 
     \[
     \frac{dy}{dx} = \frac{1}{2} \cdot (5x + 2)^{-1/2}
     \]
   - Now, multiply by the derivative of the inside function \( u(x) = 5x + 2 \), which is \( \frac{du}{dx} = 5 \).

4. Combine:
   \[
   \frac{dy}{dx} = \frac{1}{2} \cdot (5x + 2)^{-1/2} \cdot 5
   \]
   
5. Simplify:
   \[
   \frac{dy}{dx} = \frac{5}{2 \sqrt{5x + 2}}
   \]

### Final derivative:
\[
\frac{dy}{dx} = \frac{5}{2 \sqrt{5x + 2}}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you apply the chain rule to more complex functions?
2. What is the derivative of \( \sqrt{x^2 + 3x} \)?
3. Can you find the second derivative of \( y = \sqrt{5x + 2} \)?
4. How do derivatives of square root functions differ from those of polynomials?
5. How would you differentiate \( y = \frac{1}{\sqrt{5x + 2}} \)?

### Tip:  
When differentiating functions involving roots or powers, rewriting them in exponent form often makes the process much easier.

find derivatives of y = square root 5x + 2

Learn how to find the derivative of the function y = √(5x + 2) using the chain rule. This step-by-step solution explains key calculus concepts like the chain rule and power rule, suitable for students in Grades 11-12.

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