https://file.aiquickdraw.com/mathbot/file/1730443603153sku24s0r.jpg

The problem in the image is in Indonesian and translates to:

"Determine the equation of the tangent line for \( f(x) = x^2 - 1 \) at the point \( x = -1 \)."

Let's solve it step-by-step.

### Step 1: Differentiate \( f(x) \)
To find the slope of the tangent line, we need the derivative of \( f(x) = x^2 - 1 \).

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

### Step 2: Calculate the Slope at \( x = -1 \)
Now, substitute \( x = -1 \) into \( f'(x) \) to find the slope of the tangent line at that point.

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

So, the slope of the tangent line is \( -2 \).

### Step 3: Find the Point on the Curve at \( x = -1 \)
We also need the coordinates of the point on the curve where \( x = -1 \).

\[
f(-1) = (-1)^2 - 1 = 1 - 1 = 0
\]

Thus, the point of tangency is \( (-1, 0) \).

### Step 4: Use the Point-Slope Form for the Tangent Line
The equation of a line with slope \( m \) passing through a point \( (x_1, y_1) \) is given by:

\[
y - y_1 = m(x - x_1)
\]

Substituting \( m = -2 \), \( x_1 = -1 \), and \( y_1 = 0 \):

\[
y - 0 = -2(x + 1)
\]

Simplify this equation:

\[
y = -2x - 2
\]

### Final Answer
The equation of the tangent line is:

\[
y = -2x - 2
\]

Would you like further explanation on any step?

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Here are some related questions you might find useful:

1. What is the importance of finding the tangent line in calculus?
2. How do you find the derivative of higher-degree polynomial functions?
3. What are some applications of tangent lines in real-world problems?
4. How can we determine if a line is perpendicular to a curve at a given point?
5. How would the solution differ if we had a different function, such as \( f(x) = x^3 - 1 \)?

**Tip:** The derivative at a point gives the slope of the tangent line, which provides an approximation of the function’s behavior near that point.

Tentukan persamaan garis singgung f(x) = x^2 - 1 di titik x = -1.

Determine the equation of the tangent line for the function f(x) = x^2 - 1 at x = -1. This problem involves key calculus concepts, including differentiation and using the derivative to find the slope of the tangent line. Suitable for high school students in Grades 10-12.

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