Derivative

These equations relate to the motion of bodies and their rates of change, forming a foundation for classical mechanics.

DY over DX equals F of X. DY over DX equals F of X and Y. X one equals delta Y over delta X one plus X two delta Y over delta X two equals Y.

That is good.

Getting to some real physics done, Sid.

We were getting into specific types of differential equations.

I wanted to go into some problems to give us an idea of how mechanics works for various problems.

A differential equation is an equation with a function and one or more of its derivatives, like Y plus DY/DX equals five X.

Example, an equation with the function Y and its derivative DY/DX.

And the idea is this: If a measured value va-variable or process variable is rushing toward set point really fast, derivative says, "Hold up.

We're gonna overshoot.

Let's ease off a bit And it's suddenly moving away from set point really fast Derivative says, "Something just happened.

Push harder right now." So it's basically a dampener.

It, it causes a dampening effect, and it resists rapid change.

We know that this curve, like the preceding one, will start from a height b on the y-axis.

But before we draw the curve, let us find its slope by differentiating, which gives dy/dx equals a.

The slope will be constant at an angle, the tangent of which is here called a.

Let us assign to a some numerical value, let's say one-third.

Now differentiate.

If you have forgotten, turn back to page twenty-five, or rather don't turn back, but think out the differentiation.

dy/dx equals two ax.

This shows that the steepness will not be constant.