Engineering Tables/Properties of Derivatives

From Wikibooks, open books for an open world
Jump to: navigation, search
  Derivative Conditions
1  \frac{d}{dx}c = 0
2  \frac{d}{dx} (cx) = c
3
Elementary Power Rule
 \frac{d}{dx} (x^n) = n x^{n-1}
4
Sum Rule
 \frac{d}{dx} \left( f \pm g \pm h \pm \cdots \right)  = \frac{df}{dx} \pm \frac{dg}{dx} \pm \frac{dh}{dx} \pm \cdots
5
Constant Multiple Rule
 \frac{d}{dx} (cf) = c  \frac{df}{dx}
6
Product Rule
 \frac{d}{dx} (fg) = f  \frac{dg}{dx} + g \frac{df}{dx}
6
Product Rule (Extended)
 \frac{d}{dx} (fgh) = fg  \frac{dh}{dx} + fh  \frac{dg}{dx} + gh  \frac{df}{dx}
7
Quotient Rule
 \frac{d}{dx} \left( \frac{f}{g} \right) = \frac{g \frac{df}{dx} - f \frac{dg}{dx}}{g^2} g \ne 0\,
8
Chain Rule
 \frac{df}{dx} = \frac{df}{dg} \cdot \frac{dg}{dx}
9  \frac{d}{dx} \left( f^n \right) = n f^{n-1} \frac{df}{dx}
10
Reciprocal Rule
 \frac{d}{dx}\left( \frac{1}{f} \right) = -\frac{1}{f^2}\frac{df}{dx} f \ne 0\,
11
Functional Power Rule
 \frac{d}{dx} \left( f^g \right) = 
\frac{d}{dx} \left( e^{g \ln f} \right) = 
f^g \left(\frac{g}{f} \cdot \frac{df}{dx}+ \frac{dg}{dx} \ln f \right) f > 0\,
12
Logarithm Rule
 \frac{df}{dx} = f \frac{d}{dx} \left( \ln f \right) f > 0\,
13
Inverse Function Rule
 \frac{df}{dx} = \frac{1}{\frac{dx}{df}} \frac{dx}{df} \ne 0\,
Notes:
  1. f, g, h are functions of x
  2. c, n are constants.