So a derivative is like picking two points on a graph and calculating the difference in the y-values and dividing by the difference in the x-values … when the two points you pick are infinitesimally close. Up above we’ve drawn a plot of temperature T versus distance x, and we’ve shown the derivative at three points.
The way you can picture a derivative is this: If you draw a straight line that barely touches the curve at one point only, then that line is called a tangent. And the derivative at a point tells you the slope of the tangent. Where the curve is steep, the slope is high (3) and where it’s not steep slope is low (1/3). (The -ve signs are because temp goes down as x increases.)

So a derivative is like picking two points on a graph and calculating the difference in the y-values and dividing by the difference in the x-values … when the two points you pick are infinitesimally close. Up above we’ve drawn a plot of temperature T versus distance x, and we’ve shown the derivative at three points.

The way you can picture a derivative is this: If you draw a straight line that barely touches the curve at one point only, then that line is called a tangent. And the derivative at a point tells you the slope of the tangent. Where the curve is steep, the slope is high (3) and where it’s not steep slope is low (1/3). (The -ve signs are because temp goes down as x increases.)