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The image shows the beginning of a calculus problem, asking to evaluate the derivative of a function using the first principles of calculus (also known as the definition of a derivative). The function given appears to be \( f(x) = \sqrt{x} \).

The first principle of differentiation is defined as:

\[
\frac{dy}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]

For the function \( f(x) = \sqrt{x} \), this becomes:

\[
\frac{dy}{dx} = \lim_{h \to 0} \frac{\sqrt{x+h} - \sqrt{x}}{h}
\]

Would you like me to proceed with solving this step-by-step?

Here are 5 questions to further explore this topic:

1. What are the key steps involved in applying the first principles of calculus to find the derivative of a function?
2. How do you simplify expressions involving radicals when finding derivatives from first principles?
3. What is the derivative of \( f(x) = \sqrt{x} \) using the standard differentiation rules?
4. Can the first principles method be used for any function, and are there cases where it becomes complex or challenging?
5. How do the results from first principles compare to the results obtained from using derivative rules (like the power rule)?

**Tip:** When applying first principles, rationalizing the numerator is a helpful technique for dealing with square roots.

From the first principle of calculus, evaluate dy/dx for f(x) = √x.

This example demonstrates how to evaluate the derivative of f(x) = √x using the first principles of calculus. Explore the step-by-step process of applying the limit definition of a derivative, including techniques to simplify expressions with radicals. Suitable for students in Grades 11-12.

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