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 (n-1) derivative:

f ⁽ⁿ⁾(x) = [f^(n-1)(x)]'

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)

Function linear approximation

For small Δx, we can get an approximation to f(x₀+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

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: