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Differentiation or the derivative is the instantaneous rate of change of a function with respect to one of its variables. This is equivalent to finding the slope of the tangent line to the function at a point.

derivative of a curve
Derivative of Curve

lim Δx→0 (f(x0+Δx) - f(x))/Δx = df(x)/dx.

It is possible to haves differentiate the function f(x) more than once. The second-derivative is the derivative of the derivative of a function.


There are many popular notations for writing the derivative. The usefulness of each notation varies with the context and it is sometimes advantageous to use more than one notation in a given context. The most common notations for differentiation are listed below.

Leibniz's Notation

Leibniz's notation, is d dx f(x).

The second derivative d2dx2f(x)

and Higher orders of differentiation are shown as dndxn f(x)

Lagrange's Notation

Lagrange's notation is a common notation and perhaps a more convinient form for writting in HTML. It denotes the derrivative by a superscripted prime mark '.

For example, f ' for the first derivative. Higher order derivatives up to the third order, are written by adding prime-marks. Thus f '' and f ''' are written for the second and third order, respectively. Even higher orders may be represented by an arabic number in brackets. f (n) is the nth derivative.

Newton's Notation

x . for the first derivative.
x .. for the second derivative.
x... for the third derivative.

Euler's Notation

Euler's notation is represented by a capital D. For example, Dx2f(x).

General Power Rule

If we have some function of f(x) = xn where n is a integer.

dxndx = n xn-1


Differentiate, x3, 4x3, 4x2-6x + 6


3x2, 12x2, 8x - 6

Chain Rule

Given f(x) which is a function of another function g(x). Then h(x)=f(g(x)). The derivative is given by the chain rule:

h'(x) = g'(f(x)) f'(x)


Differentiate the following functions with respect to x

i) (3x5 + 9x3 + 3x2 -62)2


  1. 2 (15x4 + 27x2 + 6x) (3x5 + 9x3 + 3x2 - 62)
    90x9 + 432x7 + 126x6 + 486x5 - 1590x4 + 36x3 - 3348x2 - 744x

Product Rule

The chain rule allows us to differentiate, a product of terms that depend on the differentiation variable. For example, U(x) and V(x) are function that depend on x. Then the differential is given by:

d dx (UV) = U dVdx + V dUdx

Quotient Rule

Where we have a quotient, the rule for differentiation is

d dx (U/V) = [V dUdx - U dVdx ]/ V2

Other Functions

Differentiation of trigonometric and other functions

f(x)f '(x)
exp(ax)a exp(ax)