Derivative

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.

f'(x)=\lim_{\Delta x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}

Second derivative

The second derivative is given by:

Or simply derive the first derivative:

f''(x)=(f'(x))'

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 \left ( \frac{f(x)}{g(x)} \right )'=\frac{f'(x)g(x)-f(x)g'(x)}{g^2(x)}
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

\left ( \frac{f(x)}{g(x)} \right )'=\frac{f'(x)g(x)-f(x)g'(x)}{g^2(x)}

Derivative chain rule

f ( g(x) ) ' = f ' ( g(x) ) ∙ g' (x)

This rule can be better understood with Lagrange's notation:

\frac{df}{dx}=\frac{df}{dg}\cdot \frac{dg}{dx}

Function linear approximation

For small Δx, we can get an approximation to f(x0+Δx), when we know f(x0) and f ' (x0):

f (x0x) ≈ 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

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