# Engineering Tables/Properties of Derivatives

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.