Derivative table

Derivative means differentiation of any function. In other words calculating a slope of a function. Here we have to discuss about derivative table. Derivative table is a collection of all differentiation formulas in summarized form in a single table. It can be different table for different function. In derivative table first type is power of X, under this type we differentiate a constant term (d/dx C=0), function X (d/dx X=1) and function x^n (d/dx x^n=n *x^n-1).

Second type in derivative table is exponential and logarithmic functions. In this type we differentiate exponential function d/dx  e^x=  e^x, logarithmic function d/dx logx= 1/x and function like b^x which is equal to bxln(b). third type in derivative table is trigonometric function, its is also know as trig derivative table, in this type we differentiate sin function d/dx sinx=cosx, cosine function, tangent function, cosec function , sec function, cotangent function. By these differential formulas we can make trig derivatives table, we can also proof trig function by algebraic method, in this method we first proof the sin function then with the help of sin proof, we can proof cosine function and in last with the help of both sin and cosine function proof we can proof tangent function.

Forth type in derivative table is inverse trigonometric function, calculus part mostly used in this type. In this type we differentiate inverse sin function d/dx sin^-1x= 1/(√1-x^2), inverse cos function, d/dx cos^-1x = -1/(√1-x^2), inverse tan function d/dxtan^-1x = 1/(1+x^2), inverse sec function d/dx sec^-1x = 1/(x*√x^2 -1), inverse cot function d/dx cot ^-1x =-1/(1+x^2). Fifth and the last type in derivative table is hyperbolic function. In this type we differentiate hyperbolic sin d/dx sinhx= coshx, hyperbolic cos function d/dx coshx= sinhx, hyberbolic sec function d/dx sechx= -tanhx*sechx, hyperbolic cosec function d/dxcosechx = -cothx* cosechx, hyperbolic tan function d/dxtanhx= 1-tanh2x, hyperbolic cot function d/dxcothx= 1-coth2x.

Double integrals in polar coordinates, in double integration we have region (D) in every trigonometry problem, region D is much easier to describe in Cartesian form. But some region such as disk, ring or any portion of disk or ring, if we use Cartesian coordinate it will be very difficult to differentiate these functions. So for these functions we use polar coordinate. Suppose in any problem disk is given with radius 2 then in Cartesian it is difficult to differentiate with limit, but when we change limits in polar coordinate in θ and r  form  then it become simple and we can integrate  it easily. Double integrals over rectangles means integration of a area made by curve, which cut x axis and y axis at equal points.