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To determine the gradient \( m \) of the tangent line to the curve at \( x = 1 \), we need to calculate the slope of the tangent, which is the derivative \( f'(x) \) evaluated at \( x = 1 \).

Looking at the graph, the orange line represents the tangent to the curve \( y = f(x) \) at \( x = 1 \). The gradient of this tangent can be found by using the rise-over-run formula, that is, the change in \( y \)-coordinates divided by the change in \( x \)-coordinates between two points on the tangent line.

From the graph:
- At \( x = 0 \), the tangent has a \( y \)-coordinate of approximately \( y = -1 \).
- At \( x = 2 \), the tangent has a \( y \)-coordinate of approximately \( y = -7 \).

The slope \( m \) can be calculated as:

\[
m = \frac{\Delta y}{\Delta x} = \frac{-7 - (-1)}{2 - 0} = \frac{-6}{2} = -3
\]

Thus, the gradient \( m \) of the tangent line is \( m = -3 \).

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you interpret the geometric meaning of a negative slope in a tangent line?
2. What does the derivative of a function represent in terms of the graph?
3. How can you use limits to define the derivative at a point?
4. How would you find the equation of the tangent line at \( x = 1 \)?
5. Can you find the second derivative \( f''(x) \), and what would it tell you about the curve?

**Tip:** The tangent line to a curve at a point represents the best linear approximation to the curve at that point, and its slope is equal to the derivative of the function at that point.

The tangent to the curve with equation y = f(x) at x = 1 is drawn below. Write down the gradient m of the tangent.

Learn how to find the gradient of the tangent line to a curve at x = 1 using differentiation. The problem involves calculating the slope using the derivative of the function and understanding tangents in calculus. Suitable for students in Grades 10-12.

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