How to Calculate the Gradient of a Curved Line?

To calculate the gradient of a curved line at a specific point, you need to find the gradient of the tangent to the curve at that point. Here's how:

Steps to Calculate the Gradient

Find the derivative (dy/dx): This represents the general formula for the gradient of the curve at any point. If the curve is defined by an equation y = f(x), then you differentiate f(x) with respect to x. The video example provides: if y = x², then dy/dx = 2x.
Identify the x-coordinate: Determine the x-coordinate of the point on the curve where you want to find the gradient.
Substitute the x-coordinate into dy/dx: Plug the x-coordinate you identified in the previous step into the derivative dy/dx. The result is the gradient of the tangent line (and thus the gradient of the curve) at that specific point.

Example

Let's say you have the curve y = x², and you want to find the gradient at the point where x = 3.

dy/dx = 2x (as shown in the video excerpt).
The x-coordinate is 3.
Substitute x = 3 into dy/dx = 2x: gradient = 2 * 3 = 6.

Therefore, the gradient of the curve y = x² at the point where x = 3 is 6.

In summary, the gradient of a curve at a particular point is equivalent to the gradient of the tangent line at that point. Calculus, specifically differentiation, provides the tools to find this gradient.