The gradient of a curve is constantly changing, and so finding the gradient is not as simple as reading the x-coefficient of the equation. Instead, we can draw tangents to the curve to calculate the gradient at the point where the tangent touches the curve.
The gradient of a curves at a given point is given as the gradient of the tangent to the curve at that point
Finding the Derivative
However, drawing tangents is not very accurate when done by eye. Instead, we can use algebra to find the gradient of a curve at a given point.
This works by drawing a cord connecting two points on the curve, y = f(x). The gradient of this cord gives an estimate for the average gradient of the curve between the two points. As you can see, the closer the two points are together, the more accurate the gradient of of the cord is as an estimation. This is because the cord gets closer and closer to being parallel to the tangent.
This can be noted as a cord between points A and B, where the horizontal distance between the points (difference in x-values) is h. Therefore, the vertical distance (difference in y-values) is f(x₀+h) - f(x₀):
Since gradient is defined as change in y-value over change in x-value (rise over run), the gradient of the cord is given as:
As the value for h gets smaller, points A and B get closer together, and the gradient of the cord becomes a better estimation for the gradient of the curve at A.