ik it’s taking the derivative of a derivative but idk how you’re meant to do that
In: Mathematics
The car’s position is changing based on its speed. The speed is changing based on how far down the gas pedal is. The position of the gas pedal depends on how fast the foot is moving.
The gas pedal position is the second derivative of the position. The speed of the foot is the second derivative of the speed of the car.
Remember that the result of a derivative of a function is another function. So you can then (if your function permits it) take the derivative again:
f(x) is some differentiable function
df/dx is its derivative. Now we call this function something else:
g(x)=df/dx
g(x) is some function. Let’s assume it’s also differentiable, then we can take the derivative of g(x):
dg/dx
And if we wanted, we could cobtinue doing this ad infinitum. As long as thz function permits it. For instance all functions with an exponential part are infinitely differentiable:
f(x)=A*exp(b*x)
df/dx=b*A*exp(b*x)=b*f(x)
And so:
d^(2)/dx^(2)=
d(b*f(x))/dx=
b*df/dx=
b*b*f=b^(2)*f
Differentiated again, it becomes b^(3)*f, tgen b^(4)*f, etc.
The derivative of a function is also a function. Taking a derivative has some rules that lets you take your first function and gives you it’s derivative function. All you need to do to get a second derivative is to make that derivative function the input function.
So it’s the rate of change of the the rate at of change like others have said.
So say you’re talking about how far something has fallen. How quickly it’s fallen is the first derivative (velocity here), and how quickly THAT changes is acceleration (gravity), and in this case is the second derivative of position
Since a derivative is a rate at which something changes, the second derivative is the rate at which that rate changes.
Let’s say you had x^3 as your function. The derivative would be 3x^2, and the second derivative would be the derivative of that, which is 6x. Just follow the same rules.
Algebraically speaking, *exactly* the same as regular derivatives. You just repeat the same action with your result. This applies to third, fourth, fifth, even billionth derivative as well 🙂
**Example:**
y=3x⁴+2x³+x²+5x+7
derivative = 12x³+6x²+2x+5
second derivative = 36x²+12x+2
third derivative = 72x+12
fourth derivative = 72
fifth derivative = 0
All of these follow the exact same mathematical rule = each one is the derivative of the previous.
A derivative is just the rate of change of a function, which is another function. But since it’s a function, you can take the rate of change of that as well (second derivative of the original function).
For example, the rate of change of distance is speed. The rate of change of speed is acceleration. Therefore acceleration is the second derivative of speed.
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