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.