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 
Currently, we have around 929 calculators and conversion tables to help you "do the math" quickly in areas such as:
 Free online calculators and tools
 Free online units conversion tools
 Free online web design tools
 Free online electricity & electronics tools
 Mathematics
 Online tool
 Text tool
 PDF tool
 Code
 Ecology
 Numbers
 Algebra
 Trigonometry
 Probability & Statistics
 Calculus & analysis
 Mathematical symbols
 Derivative
 Laplace transform
 Convolution
And we are still developing more. Our goal is to become the onestop, goto site for people who need to make quick calculations or who need to find quick answer for basic conversions.
Additionally, we believe the internet should be a source of free information. Therefore, all of our tools and services are completely free, with no registration required. We coded and developed each calculator individually and put each one through strict, comprehensive testing. However, please inform us if you notice even the slightest error – your input is extremely valuable to us. While most calculators on Justfreetools.com are designed to be universally applicable for worldwide usage, some are for specific countries only.