Calculus I – Derivatives and Integrals Quick Review
This document provides a quick summary of the most important derivative and integral rules commonly used in
Calculus I.
Common Derivatives
Derivative Rule
Formula
Power Rule
d/dx(x■) = n·x■■¹
Constant Rule
d/dx(c) = 0
Product Rule
(fg)' = f'g + fg'
Quotient Rule
(f/g)' = (f'g − fg') / g²
Chain Rule
d/dx[f(g(x))] = f'(g(x))·g'(x)
e■
d/dx(e■) = e■
ln(x)
d/dx(ln(x)) = 1/x
sin(x)
d/dx(sin(x)) = cos(x)
cos(x)
d/dx(cos(x)) = −sin(x)
Common Integrals
Integral Rule
Formula
Power Rule
∫x■ dx = x■■¹/(n+1) + C
Constant Multiple
∫cf(x)dx = c∫f(x)dx
e■
∫e■ dx = e■ + C
1/x
∫1/x dx = ln|x| + C
sin(x)
∫sin(x)dx = −cos(x) + C
cos(x)
∫cos(x)dx = sin(x) + C
Quick Tips:
• Use the Chain Rule when a function is inside another function.
• Use u-substitution when the derivative of the inside function also appears.
• Use Integration by Parts for products like x·e■ or x·sin(x).