Unit 2 Summary (College Board AP® Calculus AB): Revision Note

Differentiation definition & fundamental properties summary

Key facts & definitions

• The average rate of change between $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is

  • $\frac{y_2-y_1}{x_2-x_1}$

• The instantaneous rate of change at $x=a$ is

  • $\underset{x\to a}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}$

  • $\underset{h\to 0}{\lim }\frac{f\left(a+h\right)-f\left(a\right)}{h}$

• The derivative of the function $f$ is defined by

  • $f^{\text{'}}\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$

• A differentiable function is one for which its derivative exists at each point in its domain

Key theorems

• Differentiability & continuity

  • If a function $f$ is differentiable at $x=a$, then $f$ is continuous at $x=a$

• The product rule

  • If $h\left(x\right)=f\left(x\right)·g\left(x\right)$

  • Then $h\text{'}\left(x\right)=f\left(x\right)·g\text{'}\left(x\right)+g\left(x\right)·f\text{'}\left(x\right)$

• The quotient rule

  • If $h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$

  • Then $h\text{'}\left(x\right)=\frac{g\left(x\right)·f\text{'}\left(x\right)-f\left(x\right)·g\text{'}\left(x\right)}{{\left[g\left(x\right)\right]}^2}$

Key formulas

• The derivative of a function multiplied by a constant: If $a$ is a constant

  • If $h\left(x\right)=a·f\left(x\right)$

  • Then $h\text{'}\left(x\right)=a·f\text{'}\left(x\right)$

• The derivative of the sum or difference of two functions:

  • If $h\left(x\right)=f\left(x\right)\pm g\left(x\right)$

  • Then $h\text{'}\left(x\right)=f\text{'}\left(x\right)\pm g\text{'}\left(x\right)$

• Derivatives of common functions