Original Rule 
Generalized Rule (Chain Rule) 
Comments 
d
dx 
f(x) = g(x) 

d
dx 
f(u) = g(u) 
du
dx 

General form of Chain Rule 
d
dx 
x^{n} = nx^{ n1} 

d
dx 
u^{n} = nu^{n1} 
du
dx 

Generalized Power Rule 
d
dx 
4x^{1/2} = 2x^{3/2} 

d
dx 
4u^{1/2} = 2u^{3/2} 
du
dx 

An example of the above rule 
d
dx 
sin x = cos x 

d
dx 
sin u = cos u 
du
dx 

Take me to text on trig functions! 
d
dx 
ln x 
= 
1
x 


The derivative of the natural logarithm of a quantity is the reciprocal of that quantity, times the derivative of that quantity. 
d
dx 
log_{b}(x) 
= 
1
x ln(b) 

d
dx 
log_{b}(u)
 = 
1
u ln(b) 
du
dx 


d
dx 
e^{x} 
= 
e^{x} 

d
dx 
e^{u} 
= 
e^{u} 
du
dx 

The derivative of e raised to a quantity is e raised to that quantity, times the derivative of that quantity. 
d
dx 
b^{x} 
= 
b^{x} ln(b) 

d
dx 
b^{u} 
= 
b^{u} ln(b) 
du
dx 

