Differentiation
Differentiation
The process of finding the derivative of a given function.
Derivative
The derivative gives the slope at a point on the curve of a function.
Derivative by First Principles
If lim Δx → 0 f(x + Δx) − f(x) Δx exists, then this limit is called the derivative of f at x and is denoted by f´(x) or
d dx
f(x).
Derivatives of Standard Functions
d dx
(k)		=   0 ,   where k is a constant
d dx
(x)		=   1
d dx
(xn)		=   n xn - 1
d dx
(
1 x
)		=   −
1 x2
d dx
x		=  
1 2 x
d dx
ex		=   ex
d dx
ax		=   ax ln(a)
d dx
eax		=   a eax
d dx
ln(x)		=   1 x
d dx
loga(x)		=   1 x ln(a) ,   where a > 0, a ≠ 1
Derivatives of Trigonometric Functions
d dx
sin(x)		=   cos(x)
d dx
cos(x)		=   − sin(x)
d dx
tan(x)		=   sec2(x)
d dx
cot(x)		=   − cosec2(x)
d dx
sec(x)		=   − tan(x) sec(x)
Derivatives of Inverse Trigonometric Functions
d dx
sin-1(x)		=  
1 1 − x2
d dx
cos-1(x)		=   −
1 1 − x2
d dx
tan-1(x)		=  
1 1 + x2
d dx
cot-1(x)		=   −
1 1 + x2
d dx
sec-1(x)		=  
1 x x2 − 1
d dx
cosec-1(x)		=   −
1 x x2 − 1
Derivatives of Hyperbolic Functions
d dx
sinh(x)		=   cosh(x)
d dx
cosh(x)		=   sinh(x)
d dx
tanh(x)		=   sech2(x)
d dx
coth(x)		=   − cosech2(x)
d dx
sech(x)		=   − tanh(x) sech(x)
d dx
cosech(x)		=   − coth(x) cosech(x)
Rules of Differentiation
Sum Rule
d(u + v) dx
=
du dx
+
dv dx
Difference Rule
d(u − v) dx
=
du dx
−
dv dx
Product Rule
d(ku) dx
  =   k
du dx
,   where k is a constant
d(uv) dx
  =   u
dv dx
+ v
du dx
d(uvw) dx
  =   uv
dw dx
+ vw
du dx
+ wu
dv dx
Quotient Rule
d dx
(
u v
)   =  
v
du dx
− u
dv dx
v2
,   where v ≠ 0
Reciprocal Rule
d dx
( 1 u )   =   −
1 u2
du dx
Chain Rule
dy dx
  =  
dy du
×
du dx
Inverse Function Rule
dx dy
  =  
1 dy/dx