# Thread: Why do positive gradients for tangents show a curve is increasing?

1. ## Why do positive gradients for tangents show a curve is increasing?

Imagine you have a curve and you find the derivative of that curve.

It is a known fact that if the derivative is positive in certain areas then the curve is increasing in those areas. But why is this?

The derivative shows the gradient of a tangent at a certain point. So if you find the derivative to be positive for a certain region (say x=0 to x=2), you know that all the gradients of tangents are positive in that area. But people seem to assume the curve is increasing in this area as well why is this?

Sorry if i have worded the question badly it is hard to put into words, but i was just solving questions where is asked me to find where the curve was increasing and i started to think why positive gradients-of-tangents proved that the curve was increasing

thanks

2. A curve increases when the y value of the curve increases as the x value of the curve increases.

The gradient at a point is the gradient of the tangent at that point, and this gradient can be found using:

$\displaystyle \frac{y_2 - y_1}{x_2 - x_1}$

It is positive when y2 > y1 and x2 > x1, or when y1 > y2 and x1 > x2.

The moment you have the y and x coordinates of a point greater than the y and x coordinates of another point, you have an increasing 'curve' (it can be a line).