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 (n-1) derivative:
f (n)(x) = [f (n-1)(x)]'
Example:
Find the fourth derivative of
f (x) = 2x5
f (4)(x) = [2x5]'''' = [10x4]''' = [40x3]'' = [120x2]' = 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:
3x2 + 4x.
According to the sum rule:
a = 3, b = 4
f(x) = x2 , g(x) = x
f ' (x) = 2x ,
g' (x) = 1
(3x2 + 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(x0) and f ' (x0):
f (x0+Δx) ≈ f (
x0) + f '(x0)·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 | logb(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) = x3+5x2+x+8
f ' (x) = 3x2+2·5x+1+0 = 3x2+10x+1
Example #2
f (x) = sin(3x2)
When applying the chain rule:
f ' (x) = cos(3x2) · [3x2]' = cos(3x2) · 6x
Second derivative test
When the first derivative of a function is zero at point x0.
f '(x0) = 0
Then the second derivative at point x0 , f''(x0), can indicate the type of that point:
| f ''(x0) > 0 | local minimum |
|---|---|
| f ''(x0) < 0 | local maximum |
| f ''(x0) = 0 | undetermined |