Intro to Derivatives for AP Calc AB/BC

By Serena Huang

Before we get into the formal definition of a derivative, let’s set things straight. Derivatives are a building block for mostly every topic in calculus. They tell you how things change: from the numbers on a graph, growth of a plant, or the value of your money in the bank.

Now, imagine you are driving a car, and you want to know your exact speed at a specific moment. That's where the derivative comes in. It's all about finding the speed at a single point in time, your instantaneous rate of change, or if you graphed the motion of your car as a function, how your function is changing at a specific point in time.

But how do you find a derivative?

Well, first we can approximate it.

Source: https://xaktly.com/TheDerivative.html

In basic algebra, you learned how to find the slope of a graph: change in y divided by change in x. However, this is not a good approximation of your car’s speed.But you can move x2 close to x1, making Δx smaller and a better approximation.Make Δx even smaller and you get an even better approximation. Notice how the pink line is turning into the tangent line of the curve, or a line that only touches the curve in 1 point.

Now what if Δx got so small it was 0?

Then that’s your derivative!

Source: https://www.themathpage.com/aCalc/derivative.htm

(Slope of line at point P is your derivative!)

In math, we represent a derivative of the function f(x) to be f’(x), which equals dy/dx, or a tiny change in y divided by a tiny change in x. The formal definition of a derivative is usually through a limit, as shown in the picture before.

Source: https://calcworkshop.com/derivatives/definition-of-derivative/

• h is your point that’s getting smaller and smaller, also known as an infinitesimal change in the x coordinate

• f(x+h) is the y coordinate/value of the function f(x) at the point x+h

• f(x) is the y coordinate/value of the function f(x) at the point x

• The fraction represents the rate of change over a super small interval [x, x+h], which is also the derivative.

• This limit expression essentially tells us how the function f(x) changes as we make h smaller and smaller, approaching zero. As h approaches zero, the limit represents the instantaneous rate of change or the slope of the tangent line to the curve of the function f(x) at the point x. In simpler terms, it gives us the exact rate at which the function is changing at a specific point x, which is the fundamental concept behind derivatives in calculus.