Now if you happen to be taking a calculus-based physics course, there are a few extra calculus concepts you'll need to know. Whenever we cover them in physics, we'll always give a brief refresher, but it's a good idea to understand things like derivatives and integrals before you start the course. So let's get started here.

Graphically, what the derivative represents is just the instantaneous rate of change or the slope of a tangent line at a certain point. So if I have a parabola like this, and if I wanted the slope of a tangent line that's a line that's tangent to this point, I'm basically going to draw a line that touches the graph only once. That is what a tangent line means. For example, at this point, that's what that slope looks like. At this point, the tangent line looks like this. And at this point over here, the tangent line looks a little bit different. The slopes of each one of these lines is what the derivative represents and notice how it's constantly changing throughout this graph. So that's what the derivative is.

Mathematically, what happens is that in physics, there are going to be common types of functions that you're going to see, and we're going to have to mathematically calculate the derivatives. We're going to use these following rules: for a constant function like f(x)=3, it looks like a perfectly flat line. Derivatives of constant functions are always 0, as a flat line always has a slope of 0, which is what the derivative represents. Now if you have something like a constant times an x value, like, for example, -2x, the derivative is always just whatever the coefficient is, so in this case, it's -2.

To take derivatives using the power rule: if you have a variable like x raised to some power, you drop that exponent down in front of the x, so n×x, and then subtract the exponent by 1. For example, with x2, we make it 2x. For a polynomial, such as a combination of multiple functions like x2+3x, you will take derivatives of each of the terms independently.

Now that we've reviewed derivatives, occasionally in physics, we'll also have to use integrals, though they won't be super complicated. Graphically, the integral represents the area under a curve for a specific function. For example, the integral of a line graph under the curve from 0 to 4 is everything inside this big triangle. To approximate integrals, especially when doing graphs, you can add the areas of a bunch of rectangles or squares under the curve, or count a bunch of boxes to estimate the area.

Mathematically, integrals are like the opposite of derivatives. Here's the general form of a definite integral. If you have a constant function, you attach an x to it, so 3 becomes 3x. For xn, there's a power rule for integrals: increase the power by 1, and divide by the new exponent, so x2 becomes x33. If you have a polynomial, like x2+2x, integrate each term separately.

That's just a brief review of integrals.