Derivative rules
Derivative rules and laws. Derivatives of functions table.
Derivative definition
The derivative of a function is the ratio of the difference of function value f(x) at points x+Δx and x with Δx, when Δx is infinitesimally small. The derivative is the function slope or slope of the tangent line at point x.
Second derivative
The second derivative is given by:
Or simply derive the first derivative:
Nth derivative
The nth derivative is calculated by deriving f(x) n times.
The nth derivative is equal to the derivative of the (n1) derivative:
f^{ (n)}(x) = [f^{ (n1)}(x)]'
Example:
Find the fourth derivative of
f (x) = 2x^{5}
f ^{(4)}(x) = [2x^{5}]'''' = [10x^{4}]''' = [40x^{3}]'' = [120x^{2}]' = 240x
Derivative on graph of function
The derivative of a function is the slop of the tangential line.
Derivative rules
Derivative sum rule 
( a f (x) + bg(x) ) ' = a f ' (x) + bg' (x) 
Derivative product rule 
( f (x) ∙ g(x) ) ' = f ' (x) g(x) + f (x) g' (x) 
Derivative quotient rule  
Derivative chain rule 
f ( g(x) ) ' = f ' ( g(x) ) ∙ g' (x) 
Derivative sum rule
When a and b are constants.
( a f (x) + bg(x) ) ' = a f ' (x) + bg' (x)
Example:
Find the derivative of:
3x^{2} + 4x.
According to the sum rule:
a = 3, b = 4
f(x) = x^{2 }, g(x) = x
f ' (x) = 2x^{ }, g' (x) = 1
(3x^{2} + 4x)' = 3⋅2x+4⋅1 = 6x + 4
Derivative product rule
( f (x) ∙ g(x) ) ' = f ' (x) g(x) + f (x) g' (x)
Derivative quotient rule
Derivative chain rule
f ( g(x) ) ' = f ' ( g(x) ) ∙ g' (x)
This rule can be better understood with Lagrange's notation:
Function linear approximation
For small Δx, we can get an approximation to f(x _{0}+Δx), when we know f(x_{0}) and f ' (x_{0}):
f (x_{0}+Δx) ≈ f ( x_{0}) + f '(x_{0})⋅Δx
Derivatives of functions table
Function name  Function  Derivative 

f (x) 
f '(x)  
Constant 
const 
0 
Linear 
x 
1 
Power 
x^{ a} 
a x^{ a}^{1} 
Exponential 
e^{ x} 
e^{ x} 
Exponential 
a^{ x} 
a^{ x }ln a 
Natural logarithm 
ln(x) 

Logarithm 
log_{b}(x) 

Sine 
sin x 
cos x 
Cosine 
cos x 
sin x 
Tangent 
tan x 

Arcsine 
arcsin x 

Arccosine 
arccos x 

Arctangent 
arctan x 

Hyperbolic sine 
sinh x 
cosh x 
Hyperbolic cosine 
cosh x 
sinh x 
Hyperbolic tangent 
tanh x 

Inverse hyperbolic sine 
sinh^{1} x 

Inverse hyperbolic cosine 
cosh^{1} x 

Inverse hyperbolic tangent 
tanh^{1} x 

Derivative examples
Example #1
f (x) = x^{3}+5x^{2}+x+8
f ' (x) = 3x^{2}+2⋅5x+1+0 = 3x^{2}+10x+1
Example #2
f (x) = sin(3x^{2})
When applying the chain rule:
f ' (x) = cos(3x^{2}) ⋅ [3x^{2}]' = cos(3x^{2}) ⋅ 6x
Second derivative test
When the first derivative of a function is zero at point x_{0}.
f '(x_{0}) = 0
Then the second derivative at point x_{0} , f''(x_{0}), can indicate the type of that point:
f ''(x_{0}) > 0 
local minimum 
f ''(x_{0}) < 0 
local maximum 
f ''(x_{0}) = 0 
undetermined 