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

Currently, we have around 929 calculators and conversion tables to help you "do the math" quickly in areas such as:

And we are still developing more. Our goal is to become the one-stop, go-to 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.


This website uses cookies to improve your experience, analyze traffic and display ads. Learn more