https://file.aiquickdraw.com/mathbot/file/1727407878726ovm1bxtx.jpg

We are given a piecewise function:

\[
f(x) = \begin{cases} 
2x, & \text{for } x \geq 1 \\
x^2 + 1, & \text{for } x < 1 
\end{cases}
\]

The function is continuous at \(x = 1\), and we are asked to find the left-hand derivative of \(f(x)\) at \(x = 1\).

### Steps to Find the Left-hand Derivative

The left-hand derivative at \(x = 1\) is defined as:

\[
f'_-(1) = \lim_{x \to 1^-} \frac{f(x) - f(1)}{x - 1}
\]

Here, we are only concerned with the part of the function where \(x < 1\), which is \(f(x) = x^2 + 1\). First, we compute the value of the function at \(x = 1\) to ensure continuity.

#### Continuity Check
For continuity at \(x = 1\), both sides must equal at \(x = 1\):

- From the left (for \(x < 1\)): \( f(1) = 1^2 + 1 = 2 \)
- From the right (for \(x \geq 1\)): \( f(1) = 2(1) = 2 \)

Since both give \(f(1) = 2\), the function is continuous at \(x = 1\).

#### Left-hand Derivative
Now, calculate the left-hand derivative of \(f(x)\) for \(x < 1\), using \(f(x) = x^2 + 1\):

\[
f'_-(x) = \frac{d}{dx}(x^2 + 1) = 2x
\]

Thus, the left-hand derivative at \(x = 1\) is:

\[
f'_-(1) = 2(1) = 2
\]

### Conclusion
The left-hand derivative of the function at \(x = 1\) is \(2\).

---

Do you have any questions or need further details? Here are some related questions for deeper understanding:

1. How do we check for the continuity of a piecewise function at a given point?
2. What is the geometric meaning of the left-hand derivative?
3. How would we calculate the right-hand derivative for this function?
4. Why is it important for a function to be continuous at a point before computing derivatives?
5. How does the limit definition of a derivative help in understanding this problem?

**Tip**: When working with piecewise functions, always check both continuity and differentiability at the boundary points.

Diketahui fungsi yang kontinyu di 1. Tentukan nilai dari turunan kiri fungsi f(x)?

Learn how to find the left-hand derivative of a continuous piecewise function at x = 1. This problem involves understanding continuity, limits, and derivatives. Suitable for Grades 10-12 students.

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