To find the gradient of the tangent line to the curve $y = f(x)$ at the point where $x = -1$, we will follow these steps:

1. Identify the derivative: The gradient of the tangent line at a specific point is given by the derivative of the function $f(x)$ evaluated at that point.

2. Locate the point: From the image, the tangent line touches the curve $y = f(x)$ at $x = -1$. We will estimate this value from the graph if the derivative is not directly provided.

Based on the image:

• The tangent line at $x = -1$ has a positive slope.
• The tangent appears to rise from the lower left to the upper right, indicating a positive gradient.

To determine the exact gradient from a visual approach, we estimate the rise-over-run of the tangent line at that point.

Estimating Gradient from the Graph

To estimate the slope visually:

1. Find two points on the tangent line as close to the integer coordinates as possible.
2. Calculate the slope using the formula: $\text{slope} = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}$

If you would like a more detailed step-by-step process or the precise values from the graph, please let me know!

Would you like to proceed with a visual estimation of these points from the graph or have any other questions?

Here are some additional questions related to this topic:

1. Would you like a more detailed explanation of how to calculate the derivative of a function?
2. Are you interested in understanding how to determine the equation of the tangent line using this slope?
3. Do you need a review of how the slope of a tangent relates to the derivative?
4. Should we discuss how the second derivative influences the curve's concavity at this point?
5. Are you curious about how to calculate the derivative analytically if the equation of the function $f(x)$ is known?

Tip: The derivative of a function gives you a powerful tool to understand how the function's values change with respect to its inputs—essentially, it's the slope of the curve at any given point!