Derivative (calculus)

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.

• Derivative definition
• Derivative rules
• Derivatives table
• Derivative examples

Derivative definition

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(x₀+Δx), when we know f(x₀) and f ' (x₀):

f (x₀+Δx) ≈ f (x₀) + f '(x₀)·Δx

Derivatives 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 maximum
f ''(x0) < 0	local minimum
f ''(x0) = 0	undetermined

 

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See also

• Calculus
• Laplace transform ( ℒ  )
• Calculus symbols