Derrivatives?

Question

What is the physical significance of 2nd derrivative?

1st derrivative tell the slope of the cure!! so 2nd meand it will tell the slope of slope of the curve??

Answer 1

Well...
In other words it will tell which way and how does the curve change

Answer 2

The 1st derivative tells you the slope of the line tangent to the curve at that point. That's the same thing as the rate of change of the function's value, given a small change in the independent variable.

The 2nd derivative tells you the rate at which the slope of the tangent line is changing, given a small change in the independent variable.

The 2nd derivative shows the slope of the tangent line at given points on the graph of the 1st derivative.

The 3rd derivative shows the slope of the tangent line at given points on the graph of the 2nd derivative.

You get the picture.

The function, itself, is sometimes called its own "zeroth derivative."

y = f(x) = a x^2 + b x + c

When y = 0, x is a "root" of the function f.

The first derivative for that example equation would be

dy/dx = f'(x) = 2 a x + b

When dy/dx = 0, x is a local minimum or a local maximum of the function f.

d^2y/dx^2 = f"(x) = 2a

In this case d^2y/dx^2 is a constant which is never zero. But for some other function with a variable second derivative, the point at which d^2y/dx^2 equals zero is called an inflexion point.

At an inflexion point, the curve of the function f goes from being concave up to being concave down, or vice versa. It's a "bend" in the way the function is curving.

Answer 3

The second derivative tells you the curvature of the graph.

For example: f'(x) = 0 means that the tangent is at slope 0, or that the graph is not increasing or decreasing at x.

But f''(x) = 0 means that the graph is not curving inward or outward at x.

x^3 is a pretty good example to give.
f(x) = x^3
f'(x) = 3x^2
f''(x) = 6x

For any x, f'(x) > 0 so you know it has a positive slope at x != 0.
For x > 0, f''(x) > 0, so you know it curves "outward."
For x < 0, f''(x) < 0, so you know it curves "inward."
I use those terms roughly. As others have pointed out, it'll be more obvious as you work with the second derivative. It has some handy applications in physics.

Answer 4

The second derivative gives you the concavity of a curve. Most of the time, when y"=0 you will get the location of the inflexion points.

Answer 5

It tells you how fast the slope is changing. It's the same as vel,ocity vs acceleration in a car: velocity is how fast you are going, acceleration is how fast your velocity is changing.


The second derivative also tells you the concavity of a curve, but you'll learn this soon.

Answer 6

To put it simply


1st derivative is the rate of change, 2nd derivative is the rate of change of the rate of change(how fast the change is changing).

Then the 3rd derivative is the rate of change of the rate of change of the rate of change and so on.