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Find the derivative. Step by step and use simple method. 

The given function is:

\[
y = \sqrt{5x} - 7
\]

We will find the derivative of this function, \( y' \), step by step using simple differentiation rules.

### Step 1: Rewrite the square root as a power
First, express the square root in terms of exponents:

\[
y = (5x)^{1/2} - 7
\]

### Step 2: Apply the power rule to \( (5x)^{1/2} \)
To differentiate \( (5x)^{1/2} \), we use the chain rule:
- Differentiate the outer function \( u^{1/2} \), which gives \( \frac{1}{2} u^{-1/2} \).
- Then, multiply by the derivative of the inner function \( u = 5x \), which is 5.

So, the derivative of \( (5x)^{1/2} \) is:

\[
\frac{d}{dx}\left( (5x)^{1/2} \right) = \frac{1}{2} (5x)^{-1/2} \cdot 5 = \frac{5}{2\sqrt{5x}}
\]

### Step 3: Derivative of constant \( -7 \)
The derivative of a constant like \( -7 \) is 0.

### Step 4: Combine the results
Now, combining both terms:

\[
y' = \frac{5}{2\sqrt{5x}}
\]

Thus, the derivative of \( y = \sqrt{5x} - 7 \) is:

\[
y' = \frac{5}{2\sqrt{5x}}
\]

---

Would you like further details or explanations? Here are five questions to deepen your understanding:

1. How would you apply the chain rule to other composite functions?
2. What is the significance of rewriting the square root as a power for differentiation?
3. How would the result change if the coefficient inside the square root was different?
4. Can you explain why the derivative of a constant is zero?
5. What are some other applications of the chain rule in calculus?

**Tip:** When dealing with square roots, it's often easier to express them as fractional powers for differentiation or integration.

Find the derivative of y = √5x - 7. Step by step and use simple method.

Learn how to find the derivative of y = √5x - 7 using simple differentiation techniques. This solution includes applying the power and chain rule. Suitable for students learning calculus in Grades 11-12.

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