I looked for various answers and couldn't find anything helpful.What I know is the derivative of a curve at a point is the slope of tangent line drawn to the point.Nowwhat does second derivative mean slope of the slope of line?I know it in terms of velocity,acceleration etc. but it is hard for me to get it in math.
For example: The first derivative of $x^3$ is $3x^2$ but why is it not a straight line?In my book,it is written as slope of tangent.Please help regarding all these doubts in simple terms.I'm just in high school
Best Answer
The second derivative tells you something about how the graph curves on an interval.
If the second derivative is always positive on an interval $(a,b)$ then any chord connecting two points of the graph on that interval will lie above the graph. If the second derivative is always negative on the interval $(a,b)$ then any chord connecting two points of the graph on that interval will lie below the graph.
In the graph below of $y=x(x-1)(x+1)$ the graph has a negative second derivative on the interval $(-\infty,0)$ and a positive second derivative on the interval $(0,\infty)$ so it is concave down and concave up, respectively on the two intervals.
Another way of expressing the same idea is that if a continuous second differentiable function has a positive second derivative at point $(x_0,y_0)$ then on some neighborhood of $(x_0,y_0)$ the tangent line at $(x_0,y_0)$ lies below the graph (except at the point of tangency). If the second derivative is negative at the point of tangency the tangent line lies above the graph on some neighborhood of the point of tangency (except at the point of tangency).