When you start with AP Calculus AB, it's really important to learn about different types of derivatives. Derivatives help us see how a function changes when we change its input. They show us things like speed, slope, and curves. Here are some key types of derivatives that every student should get to know:

1. Basic Derivatives:

   • Power Rule: For any function like $f(x) = x^n$, the derivative is $f'(x) = nx^{n-1}$. This means if $f(x) = x^3$, then $f'(x) = 3x^2$.
   • Sum Rule: If you have two functions, $f(x)$ and $g(x)$, then the derivative of their sum is $f'(x) + g'(x)$.
   • Constant Rule: The derivative of a constant (a number that doesn’t change) is zero. For example, if $f(x) = c$, then $f'(x) = 0$.

2. Trigonometric Derivatives:

   • Each trigonometric function has its own derivative. For example:
     • If $f(x) = \sin(x)$, then $f'(x) = \cos(x)$.
     • If $f(x) = \cos(x)$, then $f'(x) = -\sin(x)$.

3. Product and Quotient Rules:

   • When functions are multiplied, we use the Product Rule: $(fg)' = f'g + fg'$.
   • When functions are divided, we use the Quotient Rule: $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$.

4. Chain Rule:

   • This rule helps us find the derivative of functions that are made up of other functions. If you have $f(g(x))$, the derivative is $f'(g(x))g'(x)$. For example, if $h(x) = \sin(2x)$, then $h'(x) = \cos(2x) \cdot 2 = 2\cos(2x)$.

These different types of derivatives are the building blocks of calculus. By understanding and practicing these basic rules, you will be better prepared for tackling calculus problems and even more advanced math later on!